The sum of an Infinite geometric series is 4 and the sum of the cubes of the terms of the same GP is 192. The Common Ratio of the original geometric series is:
Answer & explanation
Correct answer: option 2
The sum of an Infinite geometric series = 4
a/(1-r) = 4 ........(1)
Now, when the terms are cubed,
First term is a3, and the common ratio is r3
a3/(1−r3)=192
a3=192(1−r3) .....(2)
cubing both sides of equation (1)
a3=64(1−r)3
Put value of a3 in equation 2
192(1−r3) = 64(1−r)3
64(1−r)3=192(1−r)(1+r2+r)
(1−r)2=3(1+r2+r)
(1−r)2=3(1+r2+r)
2r2+5r+2=0
Solving for r
r = -1/2
The correct answer is Option (2) → $-\frac{1}{2}$