Question

In the expansion of $(1+x{)}^{43}$ the coefficients of the $(2r+1{)}^{th}$ and the $(r+2{)}^{th}$ terms are equal; find $r$.

Answer 1

In the expansion of $(1+x{)}^{43}$

$T_{2r+1}={}^{43}{C}_{2r}{1}^{43-2r}{x}^{2r}={}^{43}{C}_{2r}{x}^{2r}$

$T_{r+2}={}^{43}{C}_{r+1}{1}^{43-(r+1)}{x}^{r+1}={}^{43}{C}_{r+1}{x}^{r+1}$

According to the question,

$\Rightarrow$ ${}^{43}{C}_{2r}={}^{43}{C}_{r+1}$

So, either $2r=r+1$ or $2r+(r+1)=43$

$\Rightarrow$ $r=1$ or $3r=42$

$\therefore$ $r=1$ or $r=14$