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The second term of a geometric series is 2/3 and it...

The second term of a geometric series is 2/3 and its sum to infinity is 3/2. Find its common ratio.

A.

1/3

B.

2

C.

4/3

D.

2/9

Correct answer is A

T2=23;S32

Tn=arn1

T2=ar=23---eqn.(i)

S=a1r=32---eqn.(ii)

= 2a = 3(1 - r)

= 2a = 3 - 3r

∴ a = 33r2

Substitute 33r2 for a in eqn.(i)

33r2×r=23

= 3r3r22=23

= 3(3r - 3r2) = -4

= 9r - 9r2 = -4

= 9r2 - 9r - 4 = 0

= 9r2 - 12r + 3r - 4 = 0

= 3r(3r - 4) + 1(3r - 4) = 0

= (3r - 4)(3r + 1) = 0

∴ r = 43or13

For a geometric series to go to infinity, the absolute value of its common ratio must be less than 1 i.e. |r| < 1.

∴ r = -1/3 (since |-1/3| < 1)