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Title: GP with S3:S6 = 64:91: Find the Common Ratio | JEE

JEE Maths question with a full step-by-step solution.

Question
The ratio of the sum of the first three terms of a G.P. to the sum of the first six terms is 64:9164:91. The common ratio of the G.P. is
A14\frac14
B34\frac34correct
C54\frac54
D74\frac74
Solution
Step 1:
S3S6=ar3−1r−1ar6−1r−1=r3−1r6−1=r3−1(r3−1)(r3+1)=1r3+1=6491.\frac{S_3}{S_6}=\frac{a\dfrac{r^3-1}{r-1}}{a\dfrac{r^6-1}{r-1}}=\frac{r^3-1}{r^6-1}=\frac{r^3-1}{(r^3-1)(r^3+1)}=\frac{1}{r^3+1}=\frac{64}{91}.
 Step 2:
r3+1=9164 ⇒ r3=2764 ⇒ r=34.r^3+1=\frac{91}{64}\ \Rightarrow\ r^3=\frac{27}{64}\ \Rightarrow\ r=\frac34.
 Correct answer: (2)
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