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SEQUENCE AND SERIES, , SEQUENCE AND SERIES, 1. DEFINITION, Sequence is a function whose domain is the set N of natural, numbers., Real Sequence : A sequence whose range is a subset of R, is called a real sequence., , (iii), , The common difference can be zero, positive or, negative., , (iv), , The sum of the two terms of an AP equidistant, from the beginning & end is constant and equal, to the sum of first & last terms., , (v), , Any term of an AP (except the first) is equal to, half the sum of terms which are equidistant from, it. an = 1/2 (an – k + an + k), k < n., , Series : If a1, a2, a3, a4, ........., an, .......... is a sequence,, then the expression, , For k = 1, an = (1/2) (an – 1 + an + 1) ;, , a1 + a2 + a3 + a4 + a5 + ........ + ......... + an + ......... is a, series., , For k = 2, an = (1/2) (an – 2 + an + 2) and so on., , A series if finite or infinite according as the number of, terms in the corresponding sequence is finite or infinite., , (vi), , tr = Sr – Sr – 1, , (vii), , If a, b, c are in AP 2 b = a + c., , Progressions : It is not necessary that the terms of a, sequence always follow a certain pattern or they are, described by some explicit formula for the nth term. Those, sequences whose terms follow certain patterns are called, progressions., , (viii) A sequence is an AP, iff its nth terms is of the form, An + B i.e., a linear expression in n. The common, difference in such a case is A i.e., the coefficient of n., , 1.1 An Arithmetic Progression (AP), AP is a sequence whose terms increase or decrease by a, fixed number. This fixed number is called the common, difference. If a is the first term & d the common difference,, then AP can be written as a nth term of this AP as, tn = a + (n – 1) d, where d = an – an – 1., The sum of the first n terms the AP is given by ;, , Sn, , n, 2a (n 1)d, 2, , n, aA ., 2, , where A is the last term., , 1.2 Geometric Progression (GP), GP is a sequence of numbers whose first term is non zero, & each of the succeeding terms is equal to the proceeding, terms multiplied by a constant. Thus in a GP the ratio of, successive terms is constant. This constant factor is called, the COMMON RATIO of the series & obtained by dividing, any term by that which immediately proceeds it. Therefore, a, ar, ar2, ar3, ar4, ........... is a GP with a as the first term &, r as common ratio., (i), , nth term = a rn – 1, , (ii), , Sum of the Ist n terms i.e. Sn =, , (iii), , Sum of an infinite GP when |r| < 1 when n o f, rn o 0 if |r| < 1 therefore, Sf =, , Properties of Arithmetic Progression, (i), , If each term of an A.P. is increased, decreased,, multiplied or divided by the same non zero, number, then the resulting sequence is also an AP., , (ii), , 3 numbers in AP are a – d, a, a + d;, 4 numbers in AP are a – 3d, a – d, a + d, a + 3d ;, 5 numbers in AP are a – 2d, a – d, a, a + d, a + 2d;, 6 numbers in AP are a – 5d, a – 3d, a – d, a + d,, a + 3d; a + 5d., , a rn 1, r 1, , ,if r z 1., , a, | r | 1 ., 1 r, , (iv), , If each term of a GP be multiplied or divided by the, same non-zero quantity, the resulting sequence is, also a GP., , (v), , Any 3 consecutive terms of a GP can be taken as, a/r, a, ar ; any 4 consecutive terms of a GP can be, taken as a/r3, a/r, ar ar3 & so on., , (vi), , If a, b, c are in GP b2 = ac.
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SEQUENCE AND SERIES, , Properties of Geometric Progressions, 1., , If all the terms of a GP be multiplied or divided, by the same non–zero constant, then it remains a, GP with the same common ratio., , Sum of n AM’s inserted between a & b is equal to n, n, , times the single AM between a & b i.e. ¦ A r, , nA where, , r 1, , 2., , The reciprocals of the terms of a given GP forms, a GP., , A is the single AM between a & b., , 3., , If each term of a GP be raised to the same power,, the resulting sequence also forms a G.P., , 2.3 Geometric Mean, , 4., , In a finite GP the product of the terms equidistant, form the beginning and the end is always same, and is equal to the product of the first and the last, term., , If a, b, c are in GP, b is the GM between a & c. b2 = ac,, , 5., , Three non–zero numbers, a, b, c are in GP, if, b2 = ac., , 6., , If the terms of a given GP are chosen at regular, intervals, then the new sequence so formed also, forms a GP., , 7., , If a1 , a2, a3, .... , an, .... is a GP of non–zero, non–negative terms, then log a 1 , log a 2 , ...., log an, .... is an AP and vice versa., , 2. MEANS, , therefore b = ac ; a > 0, c > 0., , 2.4 nGeometric Means between a & b, If a, b are two given numbers & a, G1, G2, ........, Gn, b are, in GP. Then G1, G2, G3, ............., Gn are n GMs between a, & b. G1 = a (b/a)1/n + 1 = ar, G2 = a (b/a)2/n + 1 = ar2, ............,, Gn a (b/a)n/n + 1 = arn where r = (b/a)1/ n + 1, , The product of n GMs between a & b is equal to the nth, n, , 2.1 Arithmetic Mean, If three terms are in AP then the middle term is called the, AM between the other two, so if a, b, c, are in AP, b is AM, of a & c., , power of the single GM between a & b i.e. S G r, r 1, , where G is the single GM between a & b., , 2.5, , AM for any n positive number a 1, a2, ........., a n is ;, , A, , a1 a 2 a 3 ................. a n, n, , (G) n, , Arithmetic, Geometric and Harmonic, means between two given numbers, , Let A, G and H be arithmetic, geometric and harmonic, means of two positive numbers a and b. Then,, , ab, ,G, 2, , 2ab, ab, , 2.2 nArithmetic Means between Two Numbers, , A, , If a, b are any two given numbers & a, A1, A2, ....., An, b are, in AP then A1, A2, ..... An are n AM’s between a & b., , These three means possess the following properties, , ba, 2(b a), n (b a), , A2 a , ,......,A n a , n 1, n 1, n 1, A1= a + d, A2 = a + 2d, ............, An = a + nd, where, , A1, , a, , d, , ba, n 1, , ab and H, , 1., , A>G>H, , 2., , A, G, H form a GP i.e., G2 = AH., , 3., , The equation having a and b as its roots is, x2 – 2Ax + G2 = 0, , 4., , If A, G, H are arithmetic, geometric and harmonic, means between three given numbers a, b and c, then, the equation having a, b, c as its roots is, x 3 3Ax 2 , , 3G 2, x G3, H, , 0.
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SEQUENCE AND SERIES, n, , ¦k, , n, (n 1) (2n 1) (3n 2 3n 1), 30, , 4, , k 1, , Some important properties of Arithmetic & Geometric, Means between two quantities, , 4.4, , Sum of first n odd numbers, n, , 1., , If A and G are respectively arithmetic and, geometric means between two positive quantities, a and b, then the quadratic equation having a, b, as its roots is x2 – 2Ax + G2 = 0., , 2., , If A and G be the AM and GM between two, positive numbers, then the number are, A r A2 G2 ., , 3. SIGMA NOTATIONS, , ¦, , (i), , n, , n, , ¦k, , A series each term of which is formed by multiplying the, corresponding term of an AP & GP is called the, Arithmetico-Geometric, Series., e.g., 1 + 3x + 5x2 + 7x3 +............... Here, 1, 3, 5, ........ are in, AP & 1, x, x2, x3 ......... are in GP., , Sum of n terms of an Arithmetico, , n, , r 1, , r 1, , Let Sn = a + (a + d) r + (a + 2 d) r2 + ..... +, [a + (n – 1) d] rn – 1, , n, , r 1, , (iii), , n, , k¦ a r, , r, , n2, , 5. ARITHMETICOGEOMETRIC SERIES, , ¦ a r r ¦ br, , a r r br, , ¦ka, , 1 3 ... (2n 1), , Geometric Series, , r 1, , (ii), , 2 k 1, , k 1, , 5.1, , 3.1 Theorems, n, , ¦, , then Sn, , r 1, , k k k........... n times = nk ; where k is a, , r 1, , 5.2, , n, a, dr (1 r n 1 ) a (n 1) d r, , , , r z 1., 1 r, (1 r) 2, 1 r, , Sum to Infinity, If |r| < 1 & n of, , constant., , 4. SUM TO n TERMS OF SOME, SPECIAL SEQUENCES, , then Limit r n, n of, , 0. Sf, , a, dr, , 1 r (1 r) 2, , 6. HARMONIC PROGRESSION (HP), , 4.1, , Sum of first n natural numbers, n, , ¦k, , 1 2 3 ..... n, , n n 1, , k 1, , 4.2, , 2, , n, , 2, , 12 2 2 ..... n 2, , If the sequence a1, a2, a3, ..............., an is an HP then, 1/a1, 1/a2, .........., 1/an is an AP & converse. Here we do, not have the formula for the sum of the n terms of an HP., For HP whose first terms is a & second term is b, then nth, , 4.3, , term is t n, , n n 1, , 2n 1, , 6, , k 1, , Sum of the higher powers of first n, natural numbers, , n, , k 1, , ., , Sum of the squares of first n, natural numbers, , ¦k, , ¦k, , A sequence is said to HP if the reciprocals of its terms are, in AP., , 3, , 13 23 ........ n 3, , § n n 1 ·, ¨, ¸, 2, ©, ¹, , 2, , §, ·, ¨¦k¸, ©k 1 ¹, n, , 2, , ., , ab, b (n 1)(a b), , If a, b, c are in HP b, , 2ac, a, or, ac, c, , a b, ., bc, , 7. HARMONIC MEAN, If a, b, c are in HP, b is the HM between a & c, then, b = 2ac/[a + c].]
