Solveeit Logo

Question

Question: If ![](https://cdn.pureessence.tech/canvas_674.png?top_left_x=1514&top_left_y=0&width=300&height=243...

If {(1+4x+12)n(14x+12)n}\left\{ \left( 1 + \frac{\sqrt{4x + 1}}{2} \right)^{n} - \left( 1 - \frac{\sqrt{4x + 1}}{2} \right)^{n} \right\} = a0 + a1x + ..... + a5x5, then n equals –

A

11

B

9

C

10

D

None of these

Answer

11

Explanation

Solution

14x+1\frac{1}{\sqrt{4x + 1}} {(1+4x+12)n(14x+12)n}\left\{ \left( 1 + \frac{\sqrt{4x + 1}}{2} \right)^{n} - \left( 1 - \frac{\sqrt{4x + 1}}{2} \right)^{n} \right\}

= 12n4x+1\frac{1}{2^{n}\sqrt{4x + 1}} [2{nC1 4x+1\sqrt{4x + 1} + nC3 (4x+1\sqrt{4x + 1})3 + .......}]

= 12n1\frac{1}{2^{n - 1}} [nC1 + nC3 (4x+1\sqrt{4x + 1})2 + nC5 (4x+1\sqrt{4x + 1})4 + ......]

Also, L.H.S = a0 + a1x + ...... + a5x5, so the last term must be nC11 (4x+1\sqrt{4x + 1})10.

\ The value of n is 11.