Question

The sum of first 4 terms of a G.P. is 65/12 and sum of their reciprocals is 65/18. If product of first 3 terms is 1 and the 3rd term is α, then 2α is _________

Hint:

For any G.P. with $n$ terms, $\frac{\text{Sum of terms}}{\text{Sum of reciprocals}} = \text{Product of first and last term} = a^2 r^{n-1}$.

Answer 1

Correct Answer: 3

Step 1: $S_4 = a\frac{r^4-1}{r-1} = \frac{65}{12}$. $S_{recip} = \frac{1}{a}\frac{(1/r)^4-1}{1/r-1} = \frac{1}{ar^3}\frac{r^4-1}{r-1} = \frac{65}{18}$.
Step 2: $\frac{S_4}{S_{recip}} = a^2 r^3 = \frac{18}{12} = \frac{3}{2}$.
Step 3: Product $a \cdot ar \cdot ar^2 = a^3 r^3 = 1 \Rightarrow ar = 1$.
Step 4: $a^2 r^3 = (ar)^2 \cdot r = 1^2 \cdot r = \frac{3}{2}$.
Step 5: $\alpha = 3\text{rd term} = ar^2 = (ar) \cdot r = 1 \cdot \frac{3}{2} = \frac{3}{2}$.
Step 6: $2\alpha = 3$.