Question

If S_n denote the sum of n terms of an A.P. with first term a and common difference d such that $\frac{Sx}{Skx}$ is independent of x, then

(a) d= a

(b) d = 2a

(c) a = 2d

(d) d = −a

Answer 1

Here, we are given an A.P. with a as the first term and d as the common difference. The sum of n terms of the A.P. is given by S_n.

We need to find the relation between a and d such thatis independent of

So, let us first find the values of S_x and S_kx using the following formula for the sum of n terms of an A.P.,

Where; a = first term for the given A.P.

d = common difference of the given A.P.

n = number of terms

So, we get,

Similarly,

So,

Now, to get a term independent of x we have to eliminate the other terms, so we get

So, if we substitute, we get,

Therefore,

Hence, the correct option is (b).