Question

If the coefficients of $p^{th}$, $(p+1{)}^{th}$ and $(p+2{)}^{th}$ terms in the expansion of $(1+x{)}^n$ are in A.P., then

• A. $n^2-2np+4p^2$ = 0
• B. $n^2-n(4p+1)+4p^2-2$ = 0
• C. $n^2-n(4p+1)+4p^2$ = 0
• D. None of these

Answer 1

The correct option is B

$n^2-n(4p+1)+4p^2-2$ = 0

2${}^n{c}_p=n_{c_{p-1}}+p_{c_{p+1}}\Rightarrow \frac{2}{(n-p)!P!}=\frac{1}{(n-p+1)!(p-1)!}+\frac{1}{(n-p-1)!(p+1)!}\Rightarrow \frac{2}{(n-p)p}=\frac{1}{(n-p+1)(n-p)}+\frac{1}{(p+1)p}onsimplifingweget{n}^2-n(4p+1)+4p^2-2=0$