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Chapter, , 9, , SEQUENCES AND SERIES, vNatural numbers are the product of human spirit. – DEDEKIND v, 9.1 Introduction, In mathematics, the word, “sequence” is used in much the, same way as it is in ordinary English. When we say that a, collection of objects is listed in a sequence, we usually mean, that the collection is ordered in such a way that it has an, identified first member, second member, third member and, so on. For example, population of human beings or bacteria, at different times form a sequence. The amount of money, deposited in a bank, over a number of years form a sequence., Depreciated values of certain commodity occur in a, sequence. Sequences have important applications in several, Fibonacci, (1175-1250), spheres of human activities., Sequences, following specific patterns are called progressions. In previous class,, we have studied about arithmetic progression (A.P). In this Chapter, besides discussing, more about A.P.; arithmetic mean, geometric mean, relationship between A.M., and G.M., special series in forms of sum to n terms of consecutive natural numbers,, sum to n terms of squares of natural numbers and sum to n terms of cubes of, natural numbers will also be studied., , 9.2 Sequences, Let us consider the following examples:, Assume that there is a generation gap of 30 years, we are asked to find the, number of ancestors, i.e., parents, grandparents, great grandparents, etc. that a person, might have over 300 years., Here, the total number of generations = 300 = 10, 30, , 2020-21

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178, , MATHEMATICS, , The number of person’s ancestors for the first, second, third, …, tenth generations are, 2, 4, 8, 16, 32, …, 1024. These numbers form what we call a sequence., Consider the successive quotients that we obtain in the division of 10 by 3 at, different steps of division. In this process we get 3,3.3,3.33,3.333, ... and so on. These, quotients also form a sequence. The various numbers occurring in a sequence are, called its terms. We denote the terms of a sequence by a1, a2, a3, …, an, …, etc., the, subscripts denote the position of the term. The nth term is the number at the nth position, of the sequence and is denoted by an. The nth term is also called the general term of the, sequence., Thus, the terms of the sequence of person’s ancestors mentioned above are:, a1 = 2, a2 = 4, a3 = 8, …, a10 = 1024., Similarly, in the example of successive quotients, a1 = 3, a2 = 3.3, a3 = 3.33, …, a6 = 3.33333, etc., A sequence containing finite number of terms is called a finite sequence. For, example, sequence of ancestors is a finite sequence since it contains 10 terms (a fixed, number)., A sequence is called infinite, if it is not a finite sequence. For example, the, sequence of successive quotients mentioned above is an infinite sequence, infinite in, the sense that it never ends., Often, it is possible to express the rule, which yields the various terms of a sequence, in terms of algebraic formula. Consider for instance, the sequence of even natural, numbers 2, 4, 6, …, Here, a1 = 2 = 2 × 1, a2 = 4 = 2 × 2, a3 = 6 = 2 × 3, , a4 = 8 = 2 × 4, , ...., , ...., , ...., , ...., , ...., , ...., , ...., , ...., , ...., , ...., , ...., , ...., , a23 = 46 = 2 × 23, a24 = 48 = 2 × 24, and so on., In fact, we see that the nth term of this sequence can be written as an = 2n,, where n is a natural number. Similarly, in the sequence of odd natural numbers 1,3,5, …,, the nth term is given by the formula, an = 2n – 1, where n is a natural number., In some cases, an arrangement of numbers such as 1, 1, 2, 3, 5, 8,.. has no visible, pattern, but the sequence is generated by the recurrence relation given by, a1 = a2 = 1, a3 = a1 + a2, an = an – 2 + an – 1, n > 2, This sequence is called Fibonacci sequence., , 2020-21

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SEQUENCES AND SERIES, , 179, , In the sequence of primes 2,3,5,7,…, we find that there is no formula for the nth, prime. Such sequence can only be described by verbal description., In every sequence, we should not expect that its terms will necessarily be given, by a specific formula. However, we expect a theoretical scheme or a rule for generating, the terms a1, a2, a3,…,an,… in succession., In view of the above, a sequence can be regarded as a function whose domain, is the set of natural numbers or some subset of it. Sometimes, we use the functional, notation a(n) for an., , 9.3 Series, Let a1, a2, a3,…,an, be a given sequence. Then, the expression, a1 + a2 + a3 +,…+ an + ..., is called the series associated with the given sequence .The series is finite or infinite, according as the given sequence is finite or infinite. Series are often represented in, compact form, called sigma notation, using the Greek letter ∑ (sigma) as means of, indicating the summation involved. Thus, the series a1 + a2 + a3 + ... + an is abbreviated, n, , as, , ∑ ak ., k =1, , Remark When the series is used, it refers to the indicated sum not to the sum itself., For example, 1 + 3 + 5 + 7 is a finite series with four terms. When we use the phrase, “sum of a series,” we will mean the number that results from adding the terms, the, sum of the series is 16., We now consider some examples., Example 1 Write the first three terms in each of the following sequences defined by, the following:, (i) an = 2n + 5,, , (ii) an =, , n−3, ., 4, , Solution (i) Here an = 2n + 5, Substituting n = 1, 2, 3, we get, a1 = 2(1) + 5 = 7, a2 = 9, a3 = 11, Therefore, the required terms are 7, 9 and 11., (ii) Here an =, , 1− 3, 1, 1, n−3, = − , a2 = − , a3 = 0, . Thus, a1 =, 4, 4, 2, 4, , 2020-21

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180, , MATHEMATICS, , Hence, the first three terms are – 1 , – 1 and 0., 2, 4, Example 2 What is the 20th term of the sequence defined by, an = (n – 1) (2 – n) (3 + n) ?, Solution Putting n = 20 , we obtain, a20 = (20 – 1) (2 – 20) (3 + 20), = 19 × (– 18) × (23) = – 7866., Example 3 Let the sequence an be defined as follows:, a1 = 1, an = an – 1 + 2 for n ≥ 2., Find first five terms and write corresponding series., Solution We have, a1 = 1, a2 = a1 + 2 = 1 + 2 = 3, a3 = a2 + 2 = 3 + 2 = 5,, a4 = a3 + 2 = 5 + 2 = 7, a5 = a4 + 2 = 7 + 2 = 9., Hence, the first five terms of the sequence are 1,3,5,7 and 9. The corresponding series, is 1 + 3 + 5 + 7 + 9 +..., , EXERCISE 9.1, Write the first five terms of each of the sequences in Exercises 1 to 6 whose nth, terms are:, 1., , an = n (n + 2), , 4., , an =, , 2n − 3, 6, , n, n +1, , 3. an = 2n, , an = (–1)n–1 5n+1, , 6. a n = n, , 2. an =, , 5., , n2 + 5, ., 4, , Find the indicated terms in each of the sequences in Exercises 7 to 10 whose nth, terms are:, 7., , an = 4n – 3; a17, a24, , 9. an = (–1)n – 1n3; a9, , n2, 8. an = 2n ; a7, n(n – 2), ; a20 ., 10. an =, n+3, , 2020-21

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SEQUENCES AND SERIES, , 181, , Write the first five terms of each of the sequences in Exercises 11 to 13 and obtain the, corresponding series:, 11. a1 = 3, an = 3an – 1 + 2 for all n > 1, , 12. a1 = – 1, an =, , an −1, ,n≥2, n, , 13. a1 = a2 = 2, an = an – 1–1, n > 2, 14. The Fibonacci sequence is defined by, 1 = a1 = a2 and an = an – 1 + an – 2, n > 2., Find, , an +1, an , for n = 1, 2, 3, 4, 5, , 9.4 Arithmetic Progression (A.P.), Let us recall some formulae and properties studied earlier., A sequence a1, a2, a3,…, an ,… is called arithmetic sequence or arithmetic, progression if an + 1 = an + d, n ∈ N, where a1 is called the first term and the constant, term d is called the common difference of the A.P., Let us consider an A.P. (in its standard form) with first term a and common, difference d, i.e., a, a + d, a + 2d, ..., Then the nth term (general term) of the A.P. is an = a + (n – 1) d., We can verify the following simple properties of an A.P. :, (i) If a constant is added to each term of an A.P., the resulting sequence is, also an A.P., (ii) If a constant is subtracted from each term of an A.P., the resulting, sequence is also an A.P., (iii) If each term of an A.P. is multiplied by a constant, then the resulting, sequence is also an A.P., (iv) If each term of an A.P. is divided by a non-zero constant then the, resulting sequence is also an A.P., Here, we shall use the following notations for an arithmetic progression:, a = the first term, l = the last term, d = common difference,, n = the number of terms., Sn= the sum to n terms of A.P., Let a, a + d, a + 2d, …, a + (n – 1) d be an A.P. Then, l, , = a + (n – 1) d, , 2020-21

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182, , MATHEMATICS, , Sn =, We can also write, Sn =, , n, [2a + (n −1)d ], 2, , n, [a + l ], 2, , Let us consider some examples., Example 4 In an A.P. if mth term is n and the nth term is m, where m ≠ n, find the pth, term., Solution We have am = a + (m – 1) d = n,, and, an = a + (n – 1) d = m, Solving (1) and (2), we get, (m – n) d = n – m, or d = – 1,, and, a= n+m–1, Therefore, a p = a + (p – 1)d, = n + m – 1 + ( p – 1) (–1) = n + m – p, Hence, the pth term is n + m – p., Example 5 If the sum of n terms of an A.P. is nP +, , ... (1), ... (2), ... (3), ... (4), , 1, n(n – 1)Q , where P and Q, 2, , are constants, find the common difference., Solution Let a1, a2, … an be the given A.P. Then, Sn = a1 + a2 + a3 +...+ an–1 + an = nP +, , 1, n (n – 1) Q, 2, , Therefore, S1 = a1 = P, S2 = a1 + a2 = 2P + Q, So that, a2 = S2 – S1 = P + Q, Hence, the common difference is given by d = a2 – a1 = (P + Q) – P = Q., Example 6 The sum of n terms of two arithmetic progressions are in the ratio, (3n + 8) : (7n + 15). Find the ratio of their 12th terms., Solution Let a1, a2 and d1, d2 be the first terms and common difference of the first, and second arithmetic progression, respectively. According to the given condition, we, have, , Sum to n terms of first A.P., 3n + 8, =, Sum to n termsof second A.P. 7 n + 15, , 2020-21

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SEQUENCES AND SERIES, , or, , n, [2a1 + ( n −1 )d1 ] 3n + 8, 2, =, n, 7n +15, 2, a, +, (, n, −, 1, )d, [ 2, 2], 2, , or, , 2a1 + ( n − 1) d1 3n + 8, =, 2a2 + ( n − 1) d 2 7n + 15, , Now, , a +11d1, 12th term of first A.P., = 1, th, 12 term of second A.P a2 +11d 2, , ... (1), , 2a1 + 22d1 3 × 23 + 8, =, 2a2 + 22d 2 7 × 23 + 15, Therefore, , 183, , [By putting n = 23 in (1)], , a1 + 11d1, 12th term of first A.P., 7, = th, =, a2 + 11d 2 12 term of second A.P. 16, , Hence, the required ratio is 7 : 16., Example 7 The income of a person is Rs. 3,00,000, in the first year and he receives an, increase of Rs.10,000 to his income per year for the next 19 years. Find the total, amount, he received in 20 years., Solution Here, we have an A.P. with a = 3,00,000, d = 10,000, and n = 20., Using the sum formula, we get,, , 20, [600000 + 19 × 10000] = 10 (790000) = 79,00,000., 2, Hence, the person received Rs. 79,00,000 as the total amount at the end of 20 years., S20 =, , 9.4.1 Arithmetic mean Given two numbers a and b. We can insert a number A, between them so that a, A, b is an A.P. Such a number A is called the arithmetic mean, (A.M.) of the numbers a and b. Note that, in this case, we have, A – a = b – A,, , i.e., A =, , a +b, 2, , We may also interpret the A.M. between two numbers a and b as their, average, , a +b, . For example, the A.M. of two numbers 4 and 16 is 10. We have, thus, 2, , constructed an A.P. 4, 10, 16 by inserting a number 10 between 4 and 16. The natural, , 2020-21

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184, , MATHEMATICS, , question now arises : Can we insert two or more numbers between given two numbers, so that the resulting sequence comes out to be an A.P. ? Observe that two numbers 8, and 12 can be inserted between 4 and 16 so that the resulting sequence 4, 8, 12, 16, becomes an A.P., More generally, given any two numbers a and b, we can insert as many numbers, as we like between them such that the resulting sequence is an A.P., Let A1, A2, A3, …, An be n numbers between a and b such that a, A1, A2, A3, …,, An, b is an A.P., Here, b is the (n + 2) th term, i.e., b = a + [(n + 2) – 1]d = a + (n + 1) d., , d=, , This gives, , b−a, ., n +1, , Thus, n numbers between a and b are as follows:, A1 = a + d = a +, , b−a, n +1, , A2 = a + 2d = a +, , 2(b − a ), n +1, , A3 = a + 3d = a +, , 3(b − a ), n +1, , ....., ....., , ....., ....., , ....., ....., , An = a + nd = a +, , ....., ....., , n (b − a ), n +1 ., , Example 8 Insert 6 numbers between 3 and 24 such that the resulting sequence is, an A.P., Solution Let A1, A2, A3, A4, A5 and A6 be six numbers between 3 and 24 such that, 3, A1, A2, A3, A4, A5, A6, 24 are in A.P. Here, a = 3, b = 24, n = 8., Therefore, 24 = 3 + (8 –1) d, so that d = 3., Thus, A1 = a + d = 3 + 3 = 6;, A2 = a + 2d = 3 + 2 × 3 = 9;, A3 = a + 3d = 3 + 3 × 3 = 12; A4 = a + 4d = 3 + 4 × 3 = 15;, A5 = a + 5d = 3 + 5 × 3 = 18; A6 = a + 6d = 3 + 6 × 3 = 21., Hence, six numbers between 3 and 24 are 6, 9, 12, 15, 18 and 21., , 2020-21

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SEQUENCES AND SERIES, , 185, , EXERCISE 9.2, 1. Find the sum of odd integers from 1 to 2001., 2. Find the sum of all natural numbers lying between 100 and 1000, which are, multiples of 5., 3. In an A.P., the first term is 2 and the sum of the first five terms is one-fourth of, the next five terms. Show that 20th term is –112., 4. How many terms of the A.P. – 6, −, , 5. In an A.P., if pth term is, , 11 , – 5, … are needed to give the sum –25?, 2, , 1, 1, and qth term is , prove that the sum of first pq, p, q, , 1, (pq +1), where p ≠ q., 2, If the sum of a certain number of terms of the A.P. 25, 22, 19, … is 116. Find the, last term., Find the sum to n terms of the A.P., whose kth term is 5k + 1., If the sum of n terms of an A.P. is (pn + qn2), where p and q are constants,, find the common difference., The sums of n terms of two arithmetic progressions are in the ratio, 5n + 4 : 9n + 6. Find the ratio of their 18th terms., If the sum of first p terms of an A.P. is equal to the sum of the first q terms, then, find the sum of the first (p + q) terms., Sum of the first p, q and r terms of an A.P. are a, b and c, respectively., terms is, , 6., 7., 8., 9., 10., 11., , Prove that, , a, b, c, ( q − r ) + ( r − p ) + ( p − q) = 0, p, q, r, , 12. The ratio of the sums of m and n terms of an A.P. is m2 : n2. Show that the ratio, of mth and nth term is (2m – 1) : (2n – 1)., 13. If the sum of n terms of an A.P. is 3n2 + 5n and its mth term is 164, find the value, of m., 14. Insert five numbers between 8 and 26 such that the resulting sequence is an A.P., 15. If, , a n + bn, is the A.M. between a and b, then find the value of n., a n−1 + b n−1, , 16. Between 1 and 31, m numbers have been inserted in such a way that the resulting, sequence is an A. P. and the ratio of 7th and (m – 1)th numbers is 5 : 9. Find the, value of m., , 2020-21

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186, , MATHEMATICS, , 17. A man starts repaying a loan as first instalment of Rs. 100. If he increases the, instalment by Rs 5 every month, what amount he will pay in the 30th instalment?, 18. The difference between any two consecutive interior angles of a polygon is 5°., If the smallest angle is 120° , find the number of the sides of the polygon., , 9.5 Geometric Progression (G. P.), Let us consider the following sequences:, (i) 2,4,8,16,..., (ii), , 1 –1 1 –1, , , ,, (iii) .01,.0001,.000001,..., 9 27 81 243 ..., , In each of these sequences, how their terms progress? We note that each term, except, the first progresses in a definite order., In (i), we have a1 = 2,, , a, a2, a, = 2, 3 = 2, 4 = 2 and so on., a1, a2, a3, , 1 a2, 1 a, 1 a, 1, = , 3 = , 4=, In (ii), we observe, a1 = ,, and so on., 9 a1, 3 a2, 3 a3, 3, , Similarly, state how do the terms in (iii) progress? It is observed that in each case,, every term except the first term bears a constant ratio to the term immediately preceding, 1, it. In (i), this constant ratio is 2; in (ii), it is – 3 and in (iii), the constant ratio is 0.01., Such sequences are called geometric sequence or geometric progression abbreviated, as G.P., A sequence a1, a2, a3, …, an, … is called geometric progression, if each term is, ak + 1, non-zero and, = r (constant), for k ≥ 1., ak, By letting a1 = a, we obtain a geometric progression, a, ar, ar2, ar3,…., where a, is called the first term and r is called the common ratio of the G.P. Common ratio in, 1, geometric progression (i), (ii) and (iii) above are 2, – and 0.01, respectively., 3, As in case of arithmetic progression, the problem of finding the nth term or sum of n, terms of a geometric progression containing a large number of terms would be difficult, without the use of the formulae which we shall develop in the next Section. We shall, use the following notations with these formulae:, a = the first term, r = the common ratio, l = the last term,, n = the numbers of terms,, , 2020-21

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SEQUENCES AND SERIES, , 187, , n = the numbers of terms,, Sn = the sum of first n terms., 9.5.1 General term of a G .P. Let us consider a G.P. with first non-zero term ‘a’ and, common ratio ‘r’. Write a few terms of it. The second term is obtained by multiplying, a by r, thus a2 = ar. Similarly, third term is obtained by multiplying a2 by r. Thus,, a3 = a2r = ar2, and so on., We write below these and few more terms., 1st term = a1 = a = ar1–1, 2nd term = a2 = ar = ar2–1, 3rd term = a3 = ar2 = ar3–1, 4th term = a4 = ar3 = ar4–1, 5th term = a5 = ar4 = ar5–1, Do you see a pattern? What will be 16th term?, a16 = ar16–1 = ar15, Therefore, the pattern suggests that the nth term of a G.P. is given by, , an = ar n−1 ., Thus, a, G.P. can be written as a, ar, ar2, ar3, … arn – 1; a, ar, ar2,...,arn – 1... ;according, as G.P. is finite or infinite, respectively., The series a + ar + ar2 + ... + arn–1 or a + ar + ar2 + ... + arn–1 +...are called, finite or infinite geometric series, respectively., 9.5.2. Sum to n terms of a G .P. Let the first term of a G.P. be a and the common, ratio be r. Let us denote by Sn the sum to first n terms of G.P. Then, ... (1), Sn = a + ar + ar2 +...+ arn–1, Case 1, If r = 1, we have Sn = a + a + a + ... + a (n terms) = na, If r ≠ 1, multiplying (1) by r, we have, rSn = ar + ar2 + ar3 + ... + arn, Subtracting (2) from (1), we get (1 – r) Sn = a – arn = a(1 – rn), , Case 2, , a (1 − r n ), 1− r, , ... (2), , a( r n −1), r −1, th, th, Example 9 Find the 10 and n terms of the G.P. 5, 25,125,… ., Solution Here a = 5 and r = 5. Thus, a10 = 5(5)10–1 = 5(5)9 = 510, and an = arn–1 = 5(5)n–1 = 5n ., , This gives, , Sn =, , or, , Sn =, , Example10 Which term of the G.P., 2,8,32, ... up to n terms is 131072?, Solution Let 131072 be the nth term of the given G.P. Here a = 2 and r = 4., 65536 = 4n – 1, Therefore, 131072 = an = 2(4)n – 1 or, This gives, 48 = 4n – 1., So that n – 1 = 8, i.e., n = 9. Hence, 131072 is the 9th term of the G.P., , 2020-21

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188, , MATHEMATICS, , Example11 In a G.P., the 3rd term is 24 and the 6th term is 192.Find the 10th term., Solution Here, a3 = ar = 24, 2, , ... (1), , a6 = ar = 192, and, ... (2), Dividing (2) by (1), we get r = 2. Substituting r = 2 in (1), we get a = 6., Hence, a10 = 6 (2)9 = 3072., 5, , Example12 Find the sum of first n terms and the sum of first 5 terms of the geometric, , 2 4, series 1 + + + ..., 3 9, Solution Here a = 1 and r =, , 2, . Therefore, 3, ,   2 n , 1 −   , n, , , a (1 − r n )   3   3 1−  2  ,  , =, Sn =, =, 2,   3  , 1− r, 1−, 3,   2 5 , 211, In particular, S5 = 3 1 −    = 3 ×, 243,   3  , , =, , 211, ., 81, , 3 3, Example 13 How many terms of the G.P. 3, , ,... are needed to give the, 2 4, 3069, sum 512 ?, Solution Let n be the number of terms needed. Given that a = 3, r =, Since, , Therefore, , Sn =, , a (1 – r n ), 1− r, , 3069, =, 512, , 1, ), 2n = 6  1− 1 , , n , 1,  2 , 1−, 2, , 3(1 −, , 2020-21, , 1, 3069, and Sn =, 2, 512

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SEQUENCES AND SERIES, , 189, , 1, 3069, = 1− n, 3072, 2, , or, , 1, 3069, 3, 1, 1−, =, =, n =, 2, 3072 3072 1024, 2n = 1024 = 210, which gives n = 10., , or, or, , Example 14 The sum of first three terms of a G.P. is, , 13, and their product is – 1., 12, , Find the common ratio and the terms., Solution Let, , a, , a, ar be the first three terms of the G.P. Then, r, a, 13, + ar + a =, r, 12, , ... (1), , a,   (a ) (ar ) = – 1, r, From (2), we get a3 = – 1, i.e., a = – 1 (considering only real roots), and, , ... (2), , Substituting a = –1 in (1), we have, , 1, 13, – –1 – r =, or 12r2 + 25r + 12 = 0., r, 12, 3, 4, or – ., 4, 3, 4, 3, –3, 3, 4, –4, and , – 1,, for r =, ,, Thus, the three terms of G.P. are : , – 1, for r =, 3, 4, 4, 4, 3, 3, Example15 Find the sum of the sequence 7, 77, 777, 7777, ... to n terms., This is a quadratic in r, solving, we get r = –, , Solution This is not a G.P., however, we can relate it to a G.P. by writing the terms as, Sn = 7 + 77 + 777 + 7777 + ... to n terms, =, , 7, [9 + 99 + 999 + 9999 + ...to n term], 9, , =, , 7, [(10 − 1) + (102 − 1) + (103 −1) + (104 − 1) + ...n terms], 9, , 2020-21

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190, , MATHEMATICS, , =, , 7, [(10 + 102 + 103 + ...n terms) – (1+1+1+...n terms)], 9, ,  7  10 (10n − 1), , 7  10(10n − 1), −, n, − n ., ,  = , 9  10 − 1, 9,  9, , Example 16 A person has 2 parents, 4 grandparents, 8 great grandparents, and so on., Find the number of his ancestors during the ten generations preceding his own., =, , Solution Here a = 2, r = 2 and n = 10, a (r n −1), Sn =, r −1, , Using the sum formula, , We have, S10 = 2(210 – 1) = 2046, Hence, the number of ancestors preceding the person is 2046., 9.5.3 Geometric Mean (G .M.) The geometric mean of two positive numbers a, and b is the number ab . Therefore, the geometric mean of 2 and 8 is 4. We, observe that the three numbers 2,4,8 are consecutive terms of a G.P. This leads to a, generalisation of the concept of geometric means of two numbers., Given any two positive numbers a and b, we can insert as many numbers as, we like between them to make the resulting sequence in a G.P., Let G1, G2,…, Gn be n numbers between positive numbers a and b such that, a,G1,G2,G3,…,Gn,b is a G.P. Thus, b being the (n + 2)th term,we have, 1, , b = ar, , n +1, , ,, ,  b n + 1 ., r= , a, , or, 1, , Hence, , 2,  b  n+1, n, +1, G1 = ar = a   ,, b, , , G 2 = ar 2 = a   ,, a, a, , 3, ,  b  n+1, G 3 = ar = a   ,, a, 3, , n, ,  b  n +1, G n = ar n = a  , a, , Example17 Insert three numbers between 1 and 256 so that the resulting sequence, is a G.P., Solution Let G1, G2,G3 be three numbers between 1 and 256 such that, 1, G1,G2,G3 ,256 is a G.P., , 2020-21

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SEQUENCES AND SERIES, , 191, , Therefore, 256 = r4 giving r = ± 4 (Taking real roots only), For r = 4, we have G1 = ar = 4, G2 = ar2 = 16, G3 = ar3 = 64, Similarly, for r = – 4, numbers are – 4,16 and – 64., Hence, we can insert 4, 16, 64 between 1 and 256 so that the resulting sequences are, in G.P., , 9.6 Relationship Between A.M. and G.M., Let A and G be A.M. and G.M. of two given positive real numbers a and b, respectively., Then, , A=, , a+b, and G = ab, 2, , Thus, we have, A–G=, , a +b, a + b − 2 ab, − ab =, 2, 2, , (, , a− b, , ), , 2, , ≥0, 2, From (1), we obtain the relationship A ≥ G., =, , ... (1), , Example 18 If A.M. and G.M. of two positive numbers a and b are 10 and 8,, respectively, find the numbers., , a+b, =10, 2, , Solution Given that, , A.M. =, , and, , G.M. = ab = 8, , ... (1), ... (2), , From (1) and (2), we get, a + b = 20, ... (3), ab = 64, ... (4), 2, Putting the value of a and b from (3), (4) in the identity (a – b) = (a + b)2 – 4ab,, we get, (a – b)2 = 400 – 256 = 144, or, a – b = ± 12, ... (5), Solving (3) and (5), we obtain, a = 4, b = 16 or a = 16, b = 4, Thus, the numbers a and b are 4, 16 or 16, 4 respectively., , 2020-21

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192, , MATHEMATICS, , EXERCISE 9.3, 5 5 5, , , , ..., 2 4 8, Find the 12th term of a G.P. whose 8th term is 192 and the common ratio is 2., The 5 th, 8 th and 11 th terms of a G.P. are p, q and s, respectively. Show, that q2 = ps., The 4th term of a G.P. is square of its second term, and the first term is – 3., Determine its 7th term., Which term of the following sequences:, , 1. Find the 20th and nth terms of the G.P., 2., 3., 4., 5., , (a), , 2, 2 2 , 4,... is 128 ?, , (c), , 1 1 1, 1, , , ,... is, ?, 3 9 27, 19683, , (b), , 3 ,3,3 3 ,... is729 ?, , 2, 7, , x, – are in G.P.?, 7, 2, Find the sum to indicated number of terms in each of the geometric progressions in, Exercises 7 to 10:, 7. 0.15, 0.015, 0.0015, ... 20 terms., 6. For what values of x, the numbers –, , 8., 7 , 21 , 3 7 , ... n terms., 9. 1, – a, a2, – a3, ... n terms (if a ≠ – 1)., 10. x3, x5, x7, ... n terms (if x ≠ ± 1)., 11, , 11. Evaluate, , ∑ (2 + 3k ) ., k =1, , 39, 12. The sum of first three terms of a G.P. is, and their product is 1. Find the, 10, common ratio and the terms., 13. How many terms of G.P. 3, 32, 33, … are needed to give the sum 120?, 14. The sum of first three terms of a G.P. is 16 and the sum of the next three terms is, 128. Determine the first term, the common ratio and the sum to n terms of the G.P., 15. Given a G.P. with a = 729 and 7th term 64, determine S7., 16. Find a G.P. for which sum of the first two terms is – 4 and the fifth term is, 4 times the third term., 17. If the 4th, 10th and 16th terms of a G.P. are x, y and z, respectively. Prove that x,, y, z are in G.P., , 2020-21

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SEQUENCES AND SERIES, , 193, , 18. Find the sum to n terms of the sequence, 8, 88, 888, 8888… ., 19. Find the sum of the products of the corresponding terms of the sequences 2, 4, 8,, , 1, ., 2, Show that the products of the corresponding terms of the sequences a, ar, ar2,, …arn – 1 and A, AR, AR2, … ARn – 1 form a G.P, and find the common ratio., Find four numbers forming a geometric progression in which the third term is, greater than the first term by 9, and the second term is greater than the 4th by 18., If the pth, qth and rth terms of a G.P. are a, b and c, respectively. Prove that, aq – r br – pcP – q = 1., If the first and the nth term of a G.P. are a and b, respectively, and if P is the, product of n terms, prove that P2 = (ab)n., Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from, 16, 32 and 128, 32, 8, 2,, , 20., 21., 22., 23., 24., , 1, ., rn, 25. If a, b, c and d are in G.P. show that, (a2 + b2 + c2) (b2 + c2 + d2) = (ab + bc + cd)2 ., 26. Insert two numbers between 3 and 81 so that the resulting sequence is G.P., (n + 1)th to (2n)th term is, , 27. Find the value of n so that, , a n+1 + b n+1, may be the geometric mean between, a n + bn, , a and b., 28. The sum of two numbers is 6 times their geometric mean, show that numbers, , (, , )(, , ), , are in the ratio 3 + 2 2 : 3 − 2 2 ., 29. If A and G be A.M. and G.M., respectively between two positive numbers,, prove that the numbers are A ± ( A + G )( A − G ) ., 30. The number of bacteria in a certain culture doubles every hour. If there were 30, bacteria present in the culture originally, how many bacteria will be present at the, end of 2nd hour, 4th hour and nth hour ?, 31. What will Rs 500 amounts to in 10 years after its deposit in a bank which pays, annual interest rate of 10% compounded annually?, 32. If A.M. and G.M. of roots of a quadratic equation are 8 and 5, respectively, then, obtain the quadratic equation., , 2020-21

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194, , MATHEMATICS, , 9.7 Sum to n Terms of Special Series, We shall now find the sum of first n terms of some special series, namely;, (i) 1 + 2 + 3 +… + n (sum of first n natural numbers), (ii) 12 + 22 + 32 +… + n2(sum of squares of the first n natural numbers), (iii) 13 + 23 + 33 +… + n3(sum of cubes of the first n natural numbers)., Let us take them one by one., (i) Sn=1 + 2 + 3 + … + n, then Sn =, , n (n + 1), 2, , (See Section 9.4), , (ii) Here Sn= 12 + 22 + 32 + … + n2, We consider the identity k3 – (k – 1)3 = 3k2 – 3k + 1, Putting k = 1, 2…, n successively, we obtain, 13 – 03 = 3 (1)2 – 3 (1) + 1, 23 – 13 = 3 (2)2 – 3 (2) + 1, 33 – 23 = 3(3)2 – 3 (3) + 1, ......................................., ......................................., ......................................, n3 – (n – 1)3 = 3 (n)2 – 3 (n) + 1, Adding both sides, we get, n3 – 03 = 3 (12 + 22 + 32 + ... + n2) – 3 (1 + 2 + 3 + ... + n) + n, n, , n, , k =1, , k =1, , n3 = 3 ∑ k 2 – 3 ∑ k + n, n, , By (i), we know that, , ∑ k =1 + 2 + 3 + ... + n =, k =1, , n, , Hence Sn =, , 1, , ∑ k 2 = 3 n3 +, k =1, , n (n + 1), 2, , 3n (n + 1),  1, − n  = (2n3 + 3n 2 + n), 2,  6, , n (n +1)(2n + 1), 6, (iii) Here Sn = 13 + 23 + ...+n3, We consider the identity, (k + 1)4 – k4 = 4k3 + 6k2 + 4k + 1, Putting k = 1, 2, 3… n, we get, =, , 2020-21

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SEQUENCES AND SERIES, , 24 – 14 = 4(1)3 + 6(1)2 + 4(1) + 1, 34 – 24 = 4(2)3 + 6(2)2 + 4(2) + 1, 44 – 34 = 4(3)3 + 6(3)2 + 4(3) + 1, .................................................., .................................................., .................................................., (n – 1)4 – (n – 2)4 = 4(n – 2)3 + 6(n – 2)2 + 4(n – 2) + 1, n4 – (n – 1)4 = 4(n – 1)3 + 6(n – 1)2 + 4(n – 1) + 1, (n + 1)4 – n4 = 4n3 + 6n2 + 4n + 1, Adding both sides, we get, (n + 1)4 – 14 = 4(13 + 23 + 33 +...+ n3) + 6(12 + 22 + 32 + ...+ n2) +, 4(1 + 2 + 3 +...+ n) + n, n, , n, , n, , k =1, , k =1, , k =1, , = 4 ∑ k3 + 6 ∑ k 2 + 4 ∑ k + n, , ... (1), , From parts (i) and (ii), we know that, n, , ∑k =, , k =1, , n (n + 1), 2, , n (n + 1) (2n + 1), 6, , n, , and, , ∑ k2 =, , k =1, , Putting these values in equation (1), we obtain, n, , 4 ∑ k 3 = n 4 + 4n 3 + 6n 2 + 4 n –, k =1, , or, , 6n (n + 1) (2n + 1) 4n (n + 1), –, –n, 6, 2, , 4Sn = n4 + 4n3 + 6n2 + 4n – n (2n2 + 3n + 1) – 2n (n + 1) – n, = n4 + 2n3 + n2, =, n2(n + 1)2., n 2 (n + 1) 2 [n (n + 1) ], Sn =, =, 4, 4, , 2, , Hence,, , Example 19 Find the sum to n terms of the series: 5 + 11 + 19 + 29 + 41…, Solution Let us write, or, On subtraction, we get, , Sn = 5 + 11 + 19 + 29 + ... + an–1 + an, Sn =, 5 + 11 + 19 + ... + an–2 + an–1 + an, , 2020-21, , 195

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196, , MATHEMATICS, , 0 = 5 + [6 + 8 + 10 + 12 + ...(n – 1) terms] – an, or, , an = 5 +, , (n – 1)[12 + (n − 2) × 2], 2, , = 5 + (n – 1) (n + 4) = n2 + 3n + 1, Hence, , n, , n, , n, , n, , k =1, , k =1, , k =1, , 1, , 2, 2, Sn = ∑ ak = ∑ ( k + 3k + 1) = ∑ k + 3∑ k + n, , =, , n(n + 2)(n + 4), n(n +1)(2n + 1) 3n(n + 1), +, +n =, ., 3, 6, 2, , Example 20 Find the sum to n terms of the series whose nth term is n (n+3)., Solution Given that an = n (n + 3) = n2 + 3n, Thus, the sum to n terms is given by, Sn =, , =, , n, , n, , n, , k =1, , k =1, , k =1, , ∑ ak = ∑ k 2 + 3∑ k, n (n + 1) (2n + 1) 3n (n + 1) n(n + 1)(n + 5), +, =, ., 6, 2, 3, , EXERCISE 9.4, Find the sum to n terms of each of the series in Exercises 1 to 7., 1. 1 × 2 + 2 × 3 + 3 × 4 + 4 × 5 +..., , 2. 1 × 2 × 3 + 2 × 3 × 4 + 3 × 4 × 5 + ..., , 3. 3 × 12 + 5 × 22 + 7 × 32 + ..., , 4., , 5. 52 + 62 + 72 + ... + 202, , 6. 3 × 8 + 6 × 11 + 9 × 14 + ..., , 1, 1, 1, +, +, +, 1× 2 2 × 3 3× 4 ..., , 7. 12 + (12 + 22) + (12 + 22 + 32) + ..., Find the sum to n terms of the series in Exercises 8 to 10 whose nth terms is given by, 8. n (n+1) (n+4)., , 9. n2 + 2n, , 10. (2n – 1) 2, , 2020-21

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SEQUENCES AND SERIES, , 197, , Miscellaneous Examples, Example21 If pth, qth, r th and sth terms of an A.P. are in G.P, then show that, (p – q), (q – r), (r – s) are also in G.P., Solution Here, ap = a + (p –1) d, aq = a + (q –1) d, , ... (1), ... (2), , ar = a + (r –1) d, , ... (3), , as = a + (s –1) d, Given that ap, aq, ar and as are in G.P.,, aq, , So, , ap, , =, , ... (4), , ar aq − ar q − r, =, =, aq a p − aq p − q (why ?), , ar as ar − as r − s, = =, =, aq ar aq − ar q − r, , Similarly, , (why ?), , ... (5), , ... (6), , Hence, by (5) and (6), , q – r r – s , i.e., p – q, q – r and r – s are in G.P., =, p–q q–r, 1, , 1, , 1, , Example 22 If a, b, c are in G.P. and a x = b y = c z , prove that x, y, z are in A.P., 1, , 1, , 1, , Let a x = b y = c z = k Then, a = kx , b = ky and c = kz., Since a, b, c are in G.P., therefore,, b2 = ac, Using (1) in (2), we get, k2y = kx + z, which gives 2y = x + z., Hence, x, y and z are in A.P., Solution, , ... (1), ... (2), , Example 23 If a, b, c, d and p are different real numbers such that, (a2 + b2 + c2)p2 – 2(ab + bc + cd) p + (b2 + c2 + d2) ≤ 0, then show that a, b, c and d, are in G.P., Solution Given that, (a2 + b2 + c2) p2 – 2 (ab + bc + cd) p + (b2 + c2 + d2) ≤ 0, , 2020-21, , ... (1)

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198, , MATHEMATICS, , But L.H.S., = (a2p2 – 2abp + b2) + (b2p2 – 2bcp + c2) + (c2p2 – 2cdp + d2),, which gives (ap – b)2 + (bp – c)2 + (cp – d)2 ≥ 0, (2), , ..., , Since the sum of squares of real numbers is non negative, therefore, from (1) and (2),, we have,, (ap – b)2 + (bp – c)2 + (cp – d)2 = 0, or, , ap – b = 0, bp – c = 0, cp – d = 0, , This implies that, , b c d, = = =p, a b c, , Hence a, b, c and d are in G.P., Example 24 If p,q,r are in G.P. and the equations, px2 + 2qx + r = 0 and, dx2 + 2ex + f = 0 have a common root, then show that, Solution, , d e f, , , are in A.P., p q r, , The equation px2 + 2qx + r = 0 has roots given by, x=, , −2q ± 4q 2 − 4rp, 2p, , Since p ,q, r are in G.P. q2 = pr. Thus x =, , −q, −q, but, is also root of, p, p, , dx2 + 2ex + f = 0 (Why ?). Therefore, 2, ,  −q ,  −q , d,  + 2e ,  + f = 0,,  p ,  p , or, , dq2 – 2eqp + fp2 = 0, , ... (1), , Dividing (1) by pq2 and using q2 = pr, we get, , d 2e fp, − + = 0, or, p q pr, Hence, , d e f, , , are in A.P., p q r, , 2020-21, , 2e d f, = +, q p r

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SEQUENCES AND SERIES, , 199, , Miscellaneous Exercise On Chapter 9, 1. Show that the sum of (m + n)th and (m – n)th terms of an A.P. is equal to twice, the mth term., 2. If the sum of three numbers in A.P., is 24 and their product is 440, find the, numbers., 3. Let the sum of n, 2n, 3n terms of an A.P. be S1, S2 and S3, respectively, show that, S3 = 3(S2 – S1), 4. Find the sum of all numbers between 200 and 400 which are divisible by 7., 5. Find the sum of integers from 1 to 100 that are divisible by 2 or 5., 6. Find the sum of all two digit numbers which when divided by 4, yields 1 as, remainder., 7. If f is a function satisfying f (x +y) = f(x) f(y) for all x, y ∈ N such that, n, , f(1) = 3 and, , ∑ f (x) = 120 , find the value of n., , x =1, , 8. The sum of some terms of G.P. is 315 whose first term and the common ratio are, 5 and 2, respectively. Find the last term and the number of terms., 9. The first term of a G.P. is 1. The sum of the third term and fifth term is 90., Find the common ratio of G.P., 10. The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers, in that order, we obtain an arithmetic progression. Find the numbers., 11. A G.P. consists of an even number of terms. If the sum of all the terms is 5 times, the sum of terms occupying odd places, then find its common ratio., 12. The sum of the first four terms of an A.P. is 56. The sum of the last four terms is, 112. If its first term is 11, then find the number of terms., 13. If, , a + bx b + cx c + dx, =, =, ( x ≠ 0) , then show that a, b, c and d are in G.P., a − bx b − cx c − dx, , 14. Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P., Prove that P2Rn = Sn., 15. The pth, qth and rth terms of an A.P. are a, b, c, respectively. Show that, (q – r )a + (r – p )b + (p – q )c = 0, , 1 1 1 1  1 1, 16. If a  +  ,b  +  ,c  +  are in A.P., prove that a, b, c are in A.P., b c c a a b, 17. If a, b, c, d are in G.P, prove that (an + bn), (bn + cn), (cn + dn) are in G.P., 18. If a and b are the roots of x2 – 3x + p = 0 and c, d are roots of x2 – 12x + q = 0,, where a, b, c, d form a G.P. Prove that (q + p) : (q – p) = 17:15., , 2020-21

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200, , MATHEMATICS, , 19. The ratio of the A.M. and G.M. of two positive numbers a and b, is m : n. Show, , (, , )(, , that a : b = m + m2 – n 2 : m – m2 – n 2, , )., , 1 1 1, 20. If a, b, c are in A.P.; b, c, d are in G.P. and , , are in A.P. prove that a, c, e, c d e, are in G.P., 21. Find the sum of the following series up to n terms:, (i) 5 + 55 +555 + …, (ii) .6 +. 66 +. 666+…, 22. Find the 20th term of the series 2 × 4 + 4 × 6 + 6 × 8 + ... + n terms., 23. Find the sum of the first n terms of the series: 3+ 7 +13 +21 +31 +…, 24. If S1, S2, S3 are the sum of first n natural numbers, their squares and their, cubes, respectively, show that 9 S22 = S3 (1 + 8S1)., 25. Find the sum of the following series up to n terms:, 13 13 + 23 13 + 23 + 33, +, +, + ..., 1, 1+ 3, 1+ 3 + 5, 1 × 2 2 + 2 × 32 + ... + n × (n + 1) 2 3n + 5, =, 12 × 2 + 2 2 × 3 + ... + n2 × (n + 1) 3n + 1 ., A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to, pay the balance in annual instalments of Rs 500 plus 12% interest on the unpaid, amount. How much will the tractor cost him?, Shamshad Ali buys a scooter for Rs 22000. He pays Rs 4000 cash and agrees to, pay the balance in annual instalment of Rs 1000 plus 10% interest on the unpaid, amount. How much will the scooter cost him?, A person writes a letter to four of his friends. He asks each one of them to copy, the letter and mail to four different persons with instruction that they move the, chain similarly. Assuming that the chain is not broken and that it costs 50 paise to, mail one letter. Find the amount spent on the postage when 8th set of letter is, mailed., A man deposited Rs 10000 in a bank at the rate of 5% simple interest annually., Find the amount in 15th year since he deposited the amount and also calculate the, total amount after 20 years., A manufacturer reckons that the value of a machine, which costs him Rs. 15625,, will depreciate each year by 20%. Find the estimated value at the end of 5 years., 150 workers were engaged to finish a job in a certain number of days. 4 workers, dropped out on second day, 4 more workers dropped out on third day and so on., , 26. Show that, 27., , 28., , 29., , 30., , 31., 32., , 2020-21

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SEQUENCES AND SERIES, , 201, , It took 8 more days to finish the work. Find the number of days in which the work, was completed., , Summary, , ® By a sequence, we mean an arrangement of number in definite order according, to some rule. Also, we define a sequence as a function whose domain is the, set of natural numbers or some subsets of the type {1, 2, 3, ....k}. A sequence, containing a finite number of terms is called a finite sequence. A sequence is, called infinite if it is not a finite sequence., , ® Let a1, a2, a3, ... be the sequence, then the sum expressed as a1 + a2 + a3 + ..., , is called series. A series is called finite series if it has got finite number of, terms., ® An arithmetic progression (A.P.) is a sequence in which terms increase or, decrease regularly by the same constant. This constant is called common, difference of the A.P. Usually, we denote the first term of A.P. by a, the, common difference by d and the last term by l. The general term or the nth, term of the A.P. is given by an = a + (n – 1) d., The sum Sn of the first n terms of an A.P. is given by, , n, n, Sn = 2a + ( n – 1) d  = ( a + l ) ., 2, 2, , ® The arithmetic mean A of any two numbers a and b is given by, , a+b, i.e., the, 2, , sequence a, A, b is in A.P., , ® A sequence is said to be a geometric progression or G.P., if the ratio of any, term to its preceding term is same throughout. This constant factor is called, the common ratio. Usually, we denote the first term of a G.P. by a and its, common ratio by r. The general or the nth term of G.P. is given by an= arn – 1., The sum Sn of the first n terms of G.P. is given by, , 2020-21

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202, , MATHEMATICS, , (, , ) or a (1– r ) , if r ≠ 1, , a rn – 1, Sn =, , r –1, , n, , 1–r, , ® The geometric mean (G.M.) of any two positive numbers a and b is given by, ab i.e., the sequence a, G, b is G.P., , Historical Note, Evidence is found that Babylonians, some 4000 years ago, knew of arithmetic and, geometric sequences. According to Boethius (510), arithmetic and geometric, sequences were known to early Greek writers. Among the Indian mathematician,, Aryabhatta (476) was the first to give the formula for the sum of squares and cubes, of natural numbers in his famous work Aryabhatiyam, written around, 499. He also gave the formula for finding the sum to n terms of an arithmetic, sequence starting with p th term. Noted Indian mathematicians Brahmgupta, (598), Mahavira (850) and Bhaskara (1114-1185) also considered the sum of squares, and cubes. Another specific type of sequence having important applications in, mathematics, called Fibonacci sequence, was discovered by Italian mathematician, Leonardo Fibonacci (1170-1250). Seventeenth century witnessed the classification, of series into specific forms. In 1671 James Gregory used the term infinite series in, connection with infinite sequence. It was only through the rigorous development of, algebraic and set theoretic tools that the concepts related to sequence and series, could be formulated suitably., , —v —, , 2020-21

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CBSE Class 11 Study Material,  Printable Worksheets for Class 11, NCERT Solutions for Class 11, , NCERT Solutions for class 11 Maths,  NCERT Solutions for class 11 Physics,  NCERT Solutions for class 11 Chemistry,  NCERT Solutions for class 11 Biology,  NCERT Solutions for class 11 English,  NCERT Solutions for Class 11 English, Woven Words Essay,  NCERT Solutions for Class 11 English, Woven Short Stories,  NCERT Solutions for Class 11 English, Woven Words Poetry,  NCERT Solutions for class 11 Accountancy,  NCERT Solutions for class 11 Business, Studies,  NCERT Solutions for class 11 Economics,  NCERT Solutions for class 11 Computer, Science – Python,  Class 11 Hindi Aroh (आरोह भाग 1), , ,  Class 11 Hindi Vitan (वितान भाग 1)

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 Class 11 Sanskrit,  Class 11 History,  Class 11 Geography,  Class 11 Indian Economic Development,  Class 11 Statistics for Economics,  Class 11 Political Science,  Class 11 Psychology,  Class 11 Sociology,  Class 11 Entrepreneurship,  Maths formulas for Class 11,  Hindi Grammar for Class 11,  Class 11 English Hornbill Summaries,  Class 11 English Snapshots Summaries,  CBSE Sample Papers for Class 11,  NCERT Exemplar Class 11 Maths Solutions,  NCERT Exemplar Class 11 Physics Solutions,  NCERT Exemplar Class 11, Chemistry Solutions,  NCERT Exemplar Class 11 Biology Solutions,  RD Sharma Class 11 Solutions,  CBSE Class 11 and 12 Revised Syllabus,  MCQ Questions

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 CBSE Class 11 Physics Manual,  CBSE Class 11 Chemistry Manual,  Trigonometry Formulas,  Integration Formulas,  JEE Main Study Material,  NEET Study Material,  CBSE Class 11 Notes,  Class 11 Maths Notes,  Class 11 Physics Notes,  Class 11 Chemistry Notes,  Class 11 Biology Notes,  Class 11 English Notes,  Class 11 English Woven Words Short Stories,  CBSE Class 11 English Woven Words Essay,  CBSE Class 11 English Woven Words Poetry,  CBSE Class 11 English Snapshots,  CBSE Class 11 English Hornbill,  Class 11 Business Studies Notes,  Class 11 Accountancy Notes,  Class 11 Psychology Notes,  Class 11 Entrepreneurship Notes,  Class 11 Economics Notes

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 Class 11 Indian Economic Development, Notes,  Statistics for Economics Class 11 Notes,  Class 11 Political Science Notes,  Class 11 History Notes,  Sociology Class 11 Notes,  Geography Class 11 Notes, NCERT Books for Class 11, ,  Class 11 NCERT Maths Books,  Class 11 Physics NCERT Book,  Class 11 Chemistry NCERT Book,  Class 11 Biology NCERT Book,  Class 11 Political Theory Part-I,  Class 11 NCERT Business Studies Books,  Class 11 India Constitution at Work,  NCERT Geography Book Class 11,  NCERT Class 11 History Book,  Class 11 India Economic Development,  Class 11 NCERT English Books,  NCERT Sanksrit Books Class 11,  Class 11 Computer and Communication, Technology Book,  Class 11 NCERT Accountancy Books

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 Class 11 Statistics,  Class 11 Introduction to Psychology,  Class 11 Introducing Sociology,  Class 11 Understanding Society,  Class 11 Fine Arts,  Class 11 Heritage Craft Books,  Class 11 Nai Awaz,  Class 11 Dhanak,  Class 11 The story of Graphic Design,  Class 11 Human Ecology and Family, Sciences