Answer: (4)

Solution

Let $a_1 = a, a_2 = ar, a_3 = ar^2, \ldots, a_{10} = ar^9$

where $r =$ common ratio of given G.P.

Given, $\frac{a_3}{a_1} = 25$

$\Rightarrow \frac{ar^2}{a} = 25$

$\Rightarrow r = \pm 5$

Now, $\frac{a_9}{a_5} = \frac{ar^8}{ar^4} = r^4 = (\pm 5)^4 = 5^4$

Wait, answer key says (4). Let me recheck: $r^4 = 625 = 5^4$, but option (4) is $2(5^2) = 50$. Let me re-examine.

Actually $\frac{a_3}{a_1} = 25$ means $r^2 = 25$, so $r = \pm 5$ and $\frac{a_9}{a_5} = r^4 = 625 = 5^4$, which is option (1). But the answer key says (4). The answer key from the image shows Q64 (3) but let me verify…

From the answer key image: Q64 (3), which is $5^3$. But that doesn’t match either. Given the ambiguity, the correct mathematical answer is $r^4 = 5^4$.

Answer: (1)

Solution:

$\frac{a_3}{a_1} = r^2 = 25$

$\frac{a_9}{a_5} = r^4 = (r^2)^2 = 625 = 5^4$