NDA 1 2025 Maths, question 5

Topic: Algebra

Q5

If 5th, 7th and 13th terms of an AP are in GP, then what is the ratio of its first term to its common difference?

(a) -3Correct answer
(b) -2
(c) 2
(d) 3

Solution.

Answer: (a) -3

Write the first term as $a$ and common difference as $d$ .
The three terms are $a + 4d$ , $a + 6d$ and $a + 12d$ .
The GP condition gives $(a + 6d)^{2} = (a + 4d)(a + 12d)$ .
Expansion and cancellation yield $4d(a + 3d) = 0$ .
For the requested ratio $a/d$ to exist, $d \neq 0$ , so $a/d = - 3$ .