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

The Correct Option is C

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) \(5^4.\)

Answer 2

Let the G.P. be: \( a_1, a_2 = a_1r, a_3 = a_1r^2, \ldots \)

Given: \( \frac{a_3}{a_1} = 25 \)

Since \( a_3 = a_1 r^2 \), we have:

\[ \frac{a_3}{a_1} = \frac{a_1 r^2}{a_1} = r^2 = 25 \Rightarrow r = 5 \text{ or } -5 \]

We now need to find: \[ \frac{a_9}{a_5} \]

Using general term \( a_n = a_1 r^{n-1} \):

\[ a_9 = a_1 r^8, \quad a_5 = a_1 r^4 \Rightarrow \frac{a_9}{a_5} = \frac{a_1 r^8}{a_1 r^4} = r^4 \]

We already know \( r^2 = 25 \Rightarrow r^4 = 625 = 5^4 \)

Correct answer: 5⁴