First slide

Arithmetic progression

Question

If a is the first term, d the common difference and S_k the sum to k terms of an A.P., then for $\frac{S_{kx}}{S_{x}}$ to be independent of x

Moderate

A

$a = 2 d$
B

$a = d$
C

$2 a = d$
D

None of these

Solution

We have, $\begin{array}{l} \frac{S_{kx}}{S_{x}} = \frac{\frac{kx}{2} [2 a + (kx - 1) d]}{\frac{x}{2} [2 a + (x - 1) d]} \\ = \frac{k [(2 a - d) + kxd]}{(2 a - d) + xd} \end{array}$
For $\frac{S_{kx}}{S_{x}}$ to be independent of x, 2a – d = 0 or 2a = d.

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