Question

Let S = Σ an be an infinite series such that Sn = 3 NZ (a) What are the values of an and en? 1 an n Σ», a = nes (b) What is the value of as? (c) Find a general formula for 3, (n 2 2). 11 (d) Find the sum:

Answer 1

DO Here 5 = E an and Sn = 3-4 N2 n=1 Šan? (a) What ame the values of Ea an and n=6 D=) solni As we know that + an SN = a,+92 + of first N terms of series. iie. Sum Ean = 55 n=1 3 - (5) 3 25 5 Σ an 71 25 DE) 2 Wil an art at tag =la,+92+...+98) - (9, +9+ +98)-(9, +92 +...+95) 8 Ž an. Ian n=1 n=1 Sg 55 (5-e1)-(3-(52) ろ - 4 - 3+ 4 64 25

4. म 4- 64 25 4 25 16 64-25 25x16 3D 39 95x16 - 3 400 8 39 z an an = n=6 400

(b) What is the value of as? 95 = (as +94 + ag +92 +92) - (94 + az +92 +93) Sg-54 = (3-0)-(3-1 = 4 16 4 25 = -4 4 25 25-16 100 = 9 100 95 = 9 10o

(C) find the general formula for an in 2) son an = (aita, +...tan)-( 9, +9z+ tan)-( , +92+...+an-1) an = Sn-Sn-1 an 3 * 1)-(3-1-1,) 3 4 n 2 -3+ 4 (n-12 an = 4 (n-1)2 na an = 42 4(n-192 D2 (n-1)2 an = 402 4 (52+1-20) n2 (n-1)2 an = 4n²_4ne_4+80 ne (n-1)2 An = 80-4 n2(n-132

(d) find the sum Ean n=1 son: The sium the limit of the series of its partial sem Ž an Lim no S n noi Lim. 3-4 horoo De = 3-0 - 3 2 an = 3. n=1