Question

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

• A. n²+n(4m+1)+4m²-2=0
• B. n²+n(4m+1)+4m²+2=0
• C. (n-2m)²=n+2
• D. (n+2m)²=n+2

Answer 1

The correct option is C

(n-2m)²=n+2

Coefficient of $m^{th},(m+1{)}^{th},(m+2{)}^{th}$ terms are in A.P.

⇒${}^n{C}_{m-1}$, ${}^n{C}_m$, ${}^n{C}_{m+1}$ are in A.P.

⇒2${}^n{C}_m$ = ${}^n{C}_{m-1}$ + ${}^n{C}_{m+1}$

⇒ $\frac{2(n!)}{m!}$ = $\frac{n!}{(m-1)!(n-m+1)!}$ + $\frac{n!}{(m+1)!(n-m-1)!}$

⇒2(m+1)(n-m+1)=(m+1).m + (n-m+1)(n-m)

⇒4mn = $4m^2+n-n^2+2=0$

⇒ $(n-2m{)}^2$ = n+2.