Question

The next term of the sequence `1/(1+sqrtx),  1/(1-x),  1/(1-sqrtx)` is (x ≠ 1).

Options

• `1+2sqrtx`

• `1-2sqrtx`

• `(1-2sqrtx)/(1-x)`

• `(1+2sqrtx)/(1-x)`

Answer 1

`(1+2sqrtx)/(1-x)`

Explanation:-

Given sequence is  `1/(1+sqrtx),  1/(1-x),  1/(1-sqrtx)=(1-sqrtx)/(1-x),  1/(1-x),  (1+sqrtx)/(1-x)`

We have `1/(1-x)-(1-sqrtx)/(1-x)=sqrtx/(1-x)` and

`(1+sqrtx)/(1-x)-1/(1-x)=sqrtx/(1-x)`

∴ Given sequence is an A.P. with common difference `sqrtx/(1-x)`

Hence, the next term `(t_4)=(1+sqrtx)/(1-x)+sqrtx/(1-x)=(1+2sqrtx)/(1-x)`