Question

If the coefficients of $x^9$,$x^{10},x^{11}$ in the expansion of $(1+x{)}^n$ are in arithmetic progression then $n^2-41n=$

• A. $398$
• B. $298$
• C. $-398$
• D. $198$

Answer 1

The correct option is C $-398$

$C_9^n,C_{10}^n,C_{11}^{n}areinAP$
$\Rightarrow 2C_{10}^n=C_9^n+C_{11}^n$
$\Rightarrow 2=\frac{C_9^n+C_{11}^n}{C_{10}^n}$
$\Rightarrow 2=\frac{C_9^n}{C_{10}^n}+\frac{C_{11}^n}{C_{10}^n}$
$\Rightarrow 2=\frac{10}{n-9}+\frac{n-10}{11}$
$\Rightarrow 2=\frac{110+n^2-19n+90}{11n-99}$
$\Rightarrow n^2-41n=-398$