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Sequences and Series, , A sequence is a function whose domain is the set of N nai, $y HAs Here tay H+ 18 A, , ding sequence is finite or infinite., , then the expression a, +4) +45, number of terms in the correspon, , Progressions, , The sequences whose terms follow certain patterns are, called progressions, but It is not necessary that the terms of, a sequence always follow a certain pattern or they are, , described by some explicit formula., , Arithmetic Progression (AP), , A. sequence is called an arithmetic progression if the, difference of a terny and the previous term is always same,, ie... , 4 4 = constant (= d), ¥neN, , The constant difference, generally denoted by d is called the, , common difference., OOo cc, ., (i) 1, 4,7, 10, ... is an AP whose first term is 1 and the, , common difference is 4 -1=7-4 =3., (ii) 11, 7, 3, = 1, ... is an AP whose first term is 11 and, , the common difference is 7-11=3-7=-4., , Method to Determine whether a, Sequence is an AP or not when its nth, Term is Given, (i) Obtain a,, (ii) Replace mb: ;, (iii) ae rk, a ili, , If a, ,, — 4, is independent of n, the given sequence is an AP, , otherwise it is not an AP., , tural numbers. If ay, y+ 45+ Gaye oe 1a oom isa, , series. A series is finite or infinite according a, <, , Example 1. The first four terms of the Seen, , a, =cos| —— jare, - 2, , (a) 1,0,-1/2,1/2, (b) 0,-1,0,1, (c) 1,0,-1,0, (d) None of the above, Sol. (b) Substituting n =1, 2, 3, 4 in the given expression, ,., obtain first four terms, a, = cos x/2 =0,, a, =cos 3 x/2 =0,, Example 2. Obtain the first four terms of the sequence;, whicha, =-2and a, ,, =2a, -1 3, (a) -2,-5,-7,-9, (b) -5,-7,-9,-11, (c) -2,-5,-11,-23, (d) None of the above, Sol. (c) Substituting n =1, 2,3 and 4 in the given expressio, we get the first four terms of sequence., a, =-2, a, =2a, -1=2(-2)-1=-5,, a, =2a, -1=2(-5)-1=-1}, a, = 2a, -1=2(-11)-1=-23, Therefore, the first four terms of the sequenc:, -2,-5,-11and -23. =, , a, =Cosn=-1, a, =cos2z=1
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and stim of integers between 1 and 100 divisible, , by 5, 3 Oia 5 4120-15] © 1050, ), , and sunt of integers between 1 and 100 divisible, , by 10, 8p «10-4 0-1) 10] = 550, , «Required sum « 2550 + 1050 ~ 550 = 3050, , Insertion of n Arithmetic Means, , Let Ay. Ay, Ayes Ay bem arithmetic means beeen two, quantities aand db. Then, a, vay Ay. b is an AP., , Let be the common difference of this AP. Cle, contains (+ 2) terms,, . bea (m+ 2th term so beat (n+ Nd, , , , , => vate, nel, In general mth arithmetic mean is given by, m (b= a), A, 24+ ——, , By putting m= 1,2,3,...00m We get Ay, Agyiny Ay, which are the required arithmetic means., If there is only one arithmetic mean ‘A* between « and b,, then 4, A,b are in AP. Then,, a+b, , , , Example 7. Insert three arithmetic means between 3 and 19., (a) 8, 12, 16 (b) 7, 11, 15, (c) 6, 10, 14 (d) None of these, , Sol. (b) Let Aj, Ap, Ay be 3 arithmetic’s mean between 3 and, , 19, Then, 3, Ay, Az, Ay, 19 are in AP whose common, , difference is, 19-3 a, , 341, Ay=3+d >A =7,, Az, =3+2d = A, =11,, A; =3 43d => Ay=15, Pence, the required arithmetics mean’s are 7, 11,, 15., , Example 8. if n arithmetic means are inserted between 20, and 80 such that the ratio of first mean to the last mean, is1:3, then find the value ofn., , (a) 12 (b) 13 (c) 11 (d) 14, ‘Ol. (c) Let Aj, Az, Ay, «.» Aq be n arithmetic means between, , 20 and 80 and let d be the common difference of, the AP; 20, Ay, Ap, +++, An, 80. Then,, , d= 4, , , , , , , , ¢ = 80-20 _ 60 usingd = 2-4, , n+1 n+l n+1, Now, A,=20+d = Aj=20+ 20, n+1, , -20(2+4), n+1, , ba, , NDA/NA, 4, And Ay = 20 nd, 0, + Aart e 22, nel, -20( 27), nei, 20in+ 4, Aid ogee, An 3 2040+) 5, nei, nail, _ anvt 3, ot Ane l= 3n412, nell, , Properties of Arithmetic Progressioy,, , (i) a constant is added or subtracted from em, of an AP, then the resulting sequence ig ain re, with the same common difference, a A, (ii) If each term of a given AP is multiplied or, a non-zero constant k, then the resulting, , , , , , divide, Sequien,., Mercy, 4, , also an AP with common difference kd oy 4 7, kWh,, , is the common difference of the given Ap, , (iii) A sequence is an AP if its mth term is of hie, An+ Bie. a linear expression in m The cme, difference in such a case is A ke., the coefficieny =~, , (iv) In a finite AP the sum of the terms equidistany oe, , - the beginning and end is always same and js «, hy, the sum of first and last term qual jg, he, Ay Hy Fy Hy HH FM gH .,, , (vy) If the terms of an AP are chosen at regular int, then they form an AP en, , (vi) If the given series is in AP, then inverse of that, will be in HP. Seti, , Geometric Progression (Gp), , A sequence of non-zero numbers is called a, progression, if the ratio of a term and the term Preceding », it is always a constant quantity., , The constant ratio, generally denoted by r is called g, common ratio of the GP., , i) ied S ‘, e.g., The sequence aoa gi ee GP with first tere, 4 and common ratio (- +/(%) Uae,, 3 2 3 2, , General Term of a GP, , The nth term of a GP with first term @ and common ratioy, is given by a, =ar"~!, , GP can be written as a,ar, ar,, or aar,ar’,ar’,ar',...,ar""", ..., according as it is finite or infinite., , we
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sequences and Serie,, , The rth Term from the, The mh teri from the eng, terms is ar » Where * eos, common ratio of the Gr is the firse tern Consisting of m, , . mand r is the, E poe s The third term Of GP is 4, Find the Product of its, a oe (b) 64, Sol. (d) Let a be the first term ee, the GP. ., , Ty sar 24, We have, Product of, 3 first five te,, =1,T,73T, Ts “=, =a (ar) (ar?) (ar (ar), , =a F = (ar)5 gs, =(4)° =1024, Example 10. if 1, x, YZ, 16 are in geometri progri, , then what is the value of ; —, , =x, se ee oe, ae (a) 16, , Sol. (c) "1 x,y, 2,16 are in geometric progression, Here, a=1/=16,n=5, | =ap"-! ., , W=ter4 => pa?, x=1-r=2, yst-rag, z=1-r=8, X+y¥+zZ=24+44+8=14, Example 11. If x, y,z are the pth, qth and rth terms of a, GP, then the value ofx4~" y"~P z?~4 isequalto, , (a) 0, , (b) 1, , (c) -1, , (d) None of the above, , Sol. (b) Let A be the first term and R be the common ratio of, the GP. We have,, , Tp SARP~' =x, Ty =AR™'=y and T,=AR~'=z, , =F, , Now, x = s[ARP~Y9~=Ate Re“ Mann, y’~? =fAR '}'~? = AIT P plage Me p), 2P7Fe[AR' YP~ TAP TRIP, , Therefore, x97" y'~P 20-TafAt-"*'— PtP“ 9}, , [aes iMga ree Cg =ailrs Petr =MO— 8h, , =A°R° =1, , 45, , Selection of Terms in GP, , Sometimes it is required to select a finite number of terms, , in GP. It is always convenient if we select the terms in the, following manner, , , , Common ratio _, , , , , , Sum of n Terms of a GP, , The sum of # terms of a GP with first terms ‘a’ and common, ratio ‘r’ is given by, , , , , , Stale ese for |r|<t, = l-r l-r, a rl Ir-a, , and s,+«(2=]) <4 for| r}>1, Lr) r-l, , The thing which must be noted is that the above formulae, , do not hold for r =1. For r =1, the sum of m terms of the GP, is S, =na., , = If number of terms is infinite, then sum of the terms is, a, Sa |rf<1, tan |, , 2 3, Example 12. If xa1+2(2) (2) +... where, Ly |< 2, what is y equal to?, , , , x-1 x-1 2e-2 jy 2x41, {a) — (b) a (c) = (d) er, 2 3, Sol. (c) xateds(¥) -{%) ss, ra. S ge, => = =, jee 2-y, 2, 2x-2, , => 2x-xys2 > y=, , , , Example 13. What is the value of 7°/7 78? .7°/ ..,,, upto?, , (a) tog, ($) (0), Sol. (cd) 7°” 767"., ape, , =7'=7, , (a) 7, , 6, (c) 7
Page 4 : 46, , Insertion of n Geometric Means, 4 be # geomettic means bewween two, hb. Then,, AGG, Clearly, it contains (n+ 2, of this GP, , , , Gabisa GP, terms, Let r be the common mitio, , , , be (ne 2)h term war"!, , a, , In general mth geometric mean is given by, , a,ea(#), , a, , =, , , , By putting m= 1.2,3,.....m we get G,,, Which are the required geometric means., If there is only one geometric mean ‘G’ bewween a and h,, then 4,G,b are in GP. Then,, Ge vab, , Example 14, 5 geometric means between 576 and9 are, , (a) 268,144,72,36,18 — (b) 288, 142,74,34,16, , (c) 284, 144, 74,34, 14 (d) 286, 144,76,34,16, Sol. (a) Let G,G,,G,,G,,G, be 5 geometric, , between a =576 and b =9,, , Then, 576,G,G2,G,Gs,Gs,9 is a GP with, common ratio, , Ai) la foe, , G,=ar=576x 1 =268,, , 4144,, 4, , , , means, , G, = ar? =576 x, , C= ar =576% 1 = 72,, 1, , G, =ar* =576 x — =36, , 4 = ar” = 57 “ae 3, , 1, d= Gy =ar>=576x —, an 5 =ar 576% = =8, , Hence, 288, 144, 72, 36,18 are the required, geometric means between 576 and 9., , Properties of Geometric Progression, () If all the terms of a GP be multiplied or divided by, the same non-zero constant, then it remains a GP, with same common ratio., , (ii) The reciprocals of the terms of a given GP form a GP., , (ii) If each term of a GP be raised to the same power, the, resulting sequence also forms a GP., , (iv) In a finite GP the product of the terms equidistant, from the beginning and the end is always same and is, equal to the product of the first and the last term,, , (v) Three non-zero numbers a,b,¢ are in GP, iff b*= ae., , wre, , NDA/NA, , (vi) If the terms of a given GP are chosen at ye,, , Ril,, Intervals, then the new sequence so formed it, , , , . alse,, forms a GP, (vii) Heyyy dy, is a GP of non-zero, NON-Nep ating, terms, then log a, log aa. log ayo. I an Ap an, vice-versa., , Harmonic Progression (HP), , A sequence ay. ayy a, A non-zero numbers is Called a, , Harmonic progression, if the sequence of reciprocal of there, ' |, , ' i >, numbers be, =. ee is an AP., , a my ty,, eg The sequence 42,4,4,... is a AP because 4,, & 7 3°5'7 Ne, , sequence 1,3,5,7,... is an AP., Their is no formula for finding the sum of HP Sequence, , General Term of a HP, If the sequence dy.4,, a... is a AP, then its mh term, 1, T, «=———————, 4 vn=o(- ay, a a, , Insertion of n Harmonic Means, , Let Hy, Hy,...1, be m harmonic means between wwe, , quantities a and b. Then,, 4 Ay, Ay. A,B is a FIP., , Clearly, it contains (a + 2) terms. ‘, 1, , bd ne aay ani AR, nay aie Hh, , Let the common difference of this AP is d. Then,, aoe + 2)th term = ie +(n+Id, b boa, a-b, . (n+) ab, , In general, mth harmonic mean is given by, H (n+ 1) ab, , me ma + [n=(m =)yb, By putting m=1,2,...,.", we get Hy, Hy,...,H, which an, required harmonic means., , If there is only one harmonic mean Hf between a and b, the, , 4. H,b are in HP, Then, H = 204, ath, , Example 15, Let a,b,c are in GP with l<a<be<c @, n>1 is an integer. log, n,log, n,log.n form, Sequence. This sequence is which one of the following?, , =>, , ome i2,...0, , (a) HP (b) AP, , (c) GP (d) None of these, Sol. (a) Since, a, b,c are in GP., , i b? =ac
