Question

If the coefficients of $(2r+1{)}^{th}$ term and $(r+2{)}^{th}$ term in the expansion of $(1+x{)}^{43}$ are equal. Find r.

Answer 1

$(1+x{)}^n=\sum _{k=0}^n{}^n{c}_k(x{)}^{n-k}$

Coefficient of $(t+1)th$ term in $1+x{)}^n{=}^n{c}_t$

coefficient of $(2t+1)th$ term in $(1+x{)}^{43}=43c_{2r}$

coefficient of $(t+2)th$ term in $1+x{)}^{43}=43c_{r+1}$

$43c_{2t}=43c_{r+1}$ Given

using ${}^n{c}_t{=}^n{c}_{n-t}$

$t+t+1=43$

$t=14$