Question

An A.P. whose first term is unity and in which the sum of the first half of any even number of terms to that of the second half of the same number of terms is in constant ratio, the common difference d is given by

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (B)

2

Answer 2

Let $S_{n}$denote the sum to n terms of the A.P.

According to the given condition

$\frac{S_{n}}{S_{2n} - S_{n}} = k\forall n \geq 1$.

⇒ $\frac{S_{1}}{S_{2} - S_{1}} = \frac{S_{2}}{S_{4} - S_{2}}$⇒ $S_{1}$S₄ - S₁S₂ = $S_{2}^{2}$-S₁S₂

⇒ $S_{1}S_{4} - S_{2}^{2}$⇒ $a\left\lbrack \frac{4}{2}\left( 2a + (4 - 1)d \right) \right\rbrack = (a + a + d)^{2}$

⇒ $a(4a + 6d) = (2a + d)^{2}$⇒$4a^{2} + 6ac = 4a^{2} + 4ad + d^{2}$⇒ $2ad = d^{2}$⇒ $2a = d$ As $a$ =1, we get d = 2.