Question

If $a_1, a_2, a_3, \ldots, a_{10}$ is a geometric progression and $\frac{a_3}{a_1} = 25$ then $\frac{a_9}{a_5}$ equals

• A. 3(52)
• B. 53
• C. 54
• D. 2(52)

Answer 1

Answer: (C)

54

Answer 2

We know that $\frac{a_3}{a_1} = 25$
In a geometric progression, the ratio between consecutive terms is constant,
so we can express a3 in terms of a1 and r: $a_3 = a_1 \times r^2$
Using the given information, we have: $\frac{a_3}{a_1} = r^2 = 25$

Taking the square root of both sides, we get:
$r = \sqrt{25} = 5$
Now, to find a9/a5, we can use the formula for the nth term in a geometric progression:
$a_n = a_1 \times r^{n-1}$
Substituting n = 9 and n = 5, we have:
$a_9 = a_1 \cdot r^8 \cdot a_5$
= $a_1 \cdot r^4$
Dividing both sides of the equations, we get:
$\frac{{a_9}}{{a_5}} = \frac{{a_1 \cdot r^8}}{{a_1 \cdot r^4}} = r^4$
Substituting r = 5, we have:
$\frac{{a_9}}{{a_5}} = 5^4 = 625$

Therefore, $\frac{{a_9}}{{a_5}} = 625$, which corresponds to option (3) $5^4.$