Question

If in the expansion of $(1+y{)}^n$, the coefficients of $5^{th}$, $6^{th}$ and $7^{th}$ terms are in A.P., then n is equal to

• A. 7,11
• B. 7,14
• C. 8,16
• D. None of these

Answer 1

The correct option is B

7,14

Coefficients of the 5th, 6th and 7th terms in the given expansion are ${}^n{C}_4,{}^n{C}_5$ and ${}^n{C}_6$ These coefficients are in AP

Thus, we have

${}^{2n}{C}_5={}^n{C}_4+{}^n{C}_6$

On dividing both sides by ${}^n{C}_5$, we get:

$2=\frac{{}^n{C}_4}{{}^n{C}_5}+\frac{{}^n{C}_6}{{}^n{C}_5}$

$\Rightarrow 2=\frac{5}{n-4}+\frac{n-5}{6}$

$\Rightarrow 12n-48=30+n^2-4n-5n+20$

$\Rightarrow n^2-21n+98=0$

$\Rightarrow (n-14)(n-7)=0$

$\Rightarrow n=7$ and 14