Question

If the coefficients of $5^{th},6^{th}$ and $7^{th}$ terms in the expansion of $(1+x{)}^n$ be in A.P, then n =

• A. 7 only
• B. 14 only
• C. 7 or 14
• D. None of these

Answer 1

The correct option is C 7 or 14

Coefficient of $T_5{=}^n{C}_4,T_6{=}^n{C}_5$ and $T_7{=}^n{C}_6$
According to the condition, $2^n{C}_5{=}^n{C}_4{+}^n{C}_6$
$\Rightarrow 2\left[\frac{n!}{(n-5)!5!}\right]=[\frac{n!}{(n-4)!4!}+\frac{n!}{(n-6)!6!}]$
$\Rightarrow 2\left[\frac{1}{(n-5)5}\right]=[\frac{1}{(n-4)(n-5)}+\frac{1}{6\times 5}]$
After solving, we get n = 7 or 14.