Question

known geometric sequence (an), premises infinite series is worth 6,
if the geometric sequence (an ^ 2) has an infinite series is worth 18,
then the first term of the sequence (an) is ....

a) 4
b) 3
c) 2
d) 1
e) 1/2

Answer 1

Answer :  a = 4..  Option (1)

Let r < 1

Geometric Series:     a + a r + a r² + a r³ + .... ∞ 
Given that  Sum =  a / (1 - r)  = 6    ---- (1)
=> a = 6 - 6 r         --- (2)

Geometric Series:    a² + a² r² + a² r⁴ + a² r⁶ + a ² r⁸ + ....∞
Given that  Sum =  a² / (1 - r² ) = 18     ---- (3)

Divide  (3) by (1)  to get:     a /(1+r)  = 3
=>   a = 3 + 3 r      --- (4)

Solving  (2) and  (4), we get:   a = 4   and   r = 1/3

Series:    4, 4/3, 4/9, 4/27 ....
and          16,  16/9,  16/81,  16/729 ....