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## Sequences and Series

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**Session MPTCP05**Sequences and Series**Session Objectives**• Revisit G.P. and sum of n terms of a G.P. • Sum of infinite terms of a G.P. • Geometric Mean (G.M.) and insertion of n G.M.s between two given numbers • Arithmetico-Geometric Progression (A.G.P.) - definition, nth term • Sum of n terms of an A.G.P. • Sum of infinite terms of an A.G.P. • Harmonic Progression (H.P.) - definition, nth term**Geometric Progression**_I005 A sequence is called a geometric progression (G.P.) if the ratio between any term and the previous term is constant. The constant ratio, generally denoted by r is called the common ratio. a1 = a a2 = ar a3 = ar2 a4 = ar3 First term General Term an = ar(n-1)**Problem Solving Tip**_I005 Choose Well!!!! # Terms Common ratio 3 a/r, a, ar r 4 a/r3, a/r, ar, ar3 r2 5 a/r2, a/r, a, ar, ar2 r 6 a/r5, a/r3, a/r, ar, ar3, ar5 r2**Important Properties of G.P.s**_I005 a, b, c are in G.P. b2 = ac**Sum of n Terms of a G.P.**_I006 Sn = a+ar+ar2+ar3+ . . .+ar(n-1) ………(i) Multiplying by r, we get, rSn = ar+ar2+ar3+ . . .+ar(n-1)+arn ……...(ii) Subtracting (i) from (ii), (r-1)Sn = a(rn-1)**Sum of Infinite Terms of a G.P.**_I007 Sum of n terms of a G.P.,**Illustration**_I007 **Single Geometric Mean**_I008 G is the G.M. of a and b G2 = ab**Geometric Mean – a Definition**_I008 If n terms G1, G2, G3, . . . Gn are inserted between two numbers a and b such that a, G1, G2, G3, . . . , Gn, b form a G.P., then G1, G2, G3, . . . , Gn are called geometric means (G.M.s) of a and b.**Geometric Mean – Common Ratio**_I008 Let n G.M.s be inserted between two numbers a and b The G.P. thus formed will have (n+2) terms. Let the common ratio be r Now b = ar(n+2-1) = ar(n+1)**Property of G.M.s**_I008 Let n G.M.s G1, G2, G3, . . ., Gn be inserted between a and b. Then,**A. Let the required G.M.s be G1,**G2 and G3. Common ratio r = Illustrative Problem _I008 Q. Insert 3 G.M.s between 4 and 9**Q. If the A.M. between a and b is twice as great as the**G.M., a:b is equal to (a) (b) (c) (d) Illustrative Problem _I008**Dividing by b2 and putting =r, we get,**Illustrative Problem _I008 Q. If the A.M. between a and b is twice as great as the G.M., a:b is equal to A. Given that Squaring both sides, we get,**A.**Illustrative Problem _I008 Q. If the A.M. between a and b is twice as great as the G.M., a:b is equal to Ans : (a)**Arithmetico-Geometric Progression**_I009 A sequence is called an arithmetico-geometric progression (A.G.P.) if the nth term is a product of the nth term of an A.P. and the nth term of a G.P. a1 = a a2 = (a+d)r a3 = (a+2d)r2 a4 = (a+3d)ar3 First term General Term an = {a+(n-1)d}r(n-1)**Sum of n Terms of an A.G.P.**_I010 Consider an A.G.P. with general term {a+(n-1)d}r(n-1). Let the sum of first n terms be Sn**Illustrative Problem**_I010 Q. Find the sum of the first 10 terms of the given sequence : 1, 3x, 5x2, 7x3, . . . A. Let S = 1+3x+5x2+7x3+ . . . +{1+(10-1)2}x(10-1) S = 1+3x+5x2+7x3+ . . . +19x9 xS = x+3x2+5x3+ . . . +17x9 +19x10 S-xS = 1+(2x+2x2+2x3+ . . . 2x9)-19x10**Sum of Infinite Terms of an A.G.P.**_I011 Sum of n terms of an A.G.P.,**Sum of Infinite Terms of an A.G.P.**_I011 Sum of n terms of an A.G.P.,**Q. The sum to infinity of the series**is (a) 16/35 (b) 11/8 (c) 35/16 (d) 8/6 Illustrative Problem _I011**Q. The sum to infinity of the series**is Illustrative Problem _I011 A. Let the required sum be S Ans : (c)**Q. The sum of the infinite series 1 + (1+b)r + (1+b+b2)r2 +**(1+b+b2+b3)r3 . . ., r and b being proper fractions is : Illustrative Problem _I011**Illustrative Problem**_I011 Q. The sum of the infinite series 1 + (1+b)r + (1+b+b2)r2 + (1+b+b2+b3)r3 . . . (r and b being proper fractions ) is : A. Let the required sum be S Subtracting, we have, Ans : (a)**Harmonic Progression**_I012 A sequence is called a harmonic progression (H.P.) if the reciprocals of its terms form an A.P. First term General Term**Q. The first two terms of an infinite G.P. are together**equal to 5, and every term is 3 times the sum of all the terms that follow it, the series is : Class Exercise Q1. _I007**Class Exercise Q1.**_I007 Q. The first two terms of an infinite G.P. are together equal to 5, and every term is 3 times the sum of all the terms that follow it, the series is : A. Let the first term of the G.P. be a and the common ratio be r. Given that a+ar = 5 and Now, Ans : (a)**Q. Find the value of p, if S for the G.P.**Class Exercise Q2. _I007**Q. Find the value of p, if S for the G.P.**Class Exercise Q2. _I007 A. S for the given G.P.**Class Exercise Q3.**_I008 Q. If one G.M. G and two A.M.s p and q are inserted between two quantities, show that G2 = (2p-q)(2q-p).**Common difference =**Class Exercise Q3. _I008 Q. If one G.M. G and two A.M.s p and q are inserted between two quantities, show that G2 = (2p-q)(2q-p). A. Let the two quantities be a and b. a, p, q, b are in A.P. Q.E.D.**Class Exercise Q4.**_I008 Q. n G.M.s are inserted between 16/27 and 243/16. If the ratio of the (n-1)th G.M. to the 4th G.M. is 9 : 4, find n.**Class Exercise Q4.**_I008 Q. n G.M.s are inserted between 16/27 and 243/16. If the ratio of the (n-1)th G.M. to the 4th G.M. is 9 : 4, find n. A. Common ratio Given that**Q. Find the sum of the series :**Class Exercise Q5. _I010**Q. Find the sum of the series :**Class Exercise Q5. _I010 A. We see that**Q. Find the sum of the series :**Class Exercise Q5. _I010**Class Exercise Q6.**_I010 Q. Find sum to n terms of the series : 1+2x+3x2+4x3+ . . . (x 1)**Class Exercise Q6.**_I010 Q. Find sum to n terms of the series : 1+2x+3x2+4x3+ . . . (x 1) A. We see that an = nxn-1 Sn = 1+2x+3x2+4x3+ . . . +nxn-1 xSn = x+2x2+3x3+. . . +(n-1)xn-1+nxn (1-x)Sn = 1+(x+x2+x3+ . . . xn-1)-nxn**Q. Find the sum of infinite terms of the series :**Class Exercise Q7. _I011**Q. Find the sum of infinite terms of the series :**A. Class Exercise Q7. _I011**Q. Find the sum of the series :**Class Exercise Q8. _I011**Q. Find the sum of the series :**A. Class Exercise Q8. _I011**Class Exercise Q9.**_I012 Q. If the pth, qth and rth terms of an H.P. be a, b, c respectively, then (q-r)bc+(r-p)ca+(p-q)ab is equal to (a) 1 (b) -1 (c) 0 (d) None of these**Class Exercise Q9.**_I012 Q. If the pth, qth and rth terms of an H.P. be a, b, c respectively, then (q-r)bc+(r-p)ca+(p-q)ab is equal to (a) 1 (b) -1 (c) 0 (d) None of these A. The reciprocals the terms of the H.P. will be in A.P. Let this A.P. have first term and common difference . Given that**Class Exercise Q9.**_I012 Q. If the pth, qth and rth terms of an H.P. be a, b, c respectively, then (q-r)bc+(r-p)ca+(p-q)ab is equal to (a) 1 (b) -1 (c) 0 (d) None of these A. Taking reciprocal of (i), (ii) and (iii), we have**Class Exercise Q9**_I012 Q. If the pth, qth and rth terms of an H.P. be a, b, c respectively, then (q-r)bc+(r-p)ca+(p-q)ab is equal to (a) 1 (b) -1 (c) 0 (d) None of these A. (iv)-(v), (v)-(vi), (vi)-(iv) gives,**Class Exercise Q9**_I012 Q. If the pth, qth and rth terms of an H.P. be a, b, c respectively, then (q-r)bc+(r-p)ca+(p-q)ab is equal to (a) 1 (b) -1 (c) 0 (d) None of these A. (vii)c, (viii)a and (ix)b gives, Adding,**Class Exercise Q9**_I012 Q. If the pth, qth and rth terms of an H.P. be a, b, c respectively, then (q-r)bc+(r-p)ca+(p-q)ab is equal to (a) 1 (b) -1 (c) 0 (d) None of these A. (q-r)bc+(r-p)ca+(p-q)ab = 0 Ans : (c)**Class Exercise Q10.**_I012 Q. If ax = by = cz and x, y, z are in H.P. then a, b, c are in (a) A.P. (b) H.P. (c) G.P. (d) None of these