Question

Find the value of r if cofficient of $(2r+2{)}^{th}$ and $(r+1{)}^{th}$ terms in the expansion of $(1+x{)}^{43}$ are equal.

Answer 1

If coefficient of $(2r+2{)}^{\text{th}}$ term and $(r+1{)}^{\text{th}}$ term in $(1+x{)}^{43}$ are equal

$T_{r+1}={}^{43}{C}_r(x{)}^r$

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

Now ${}^{43}{C}_r={}^{43}{C}_{2r+1}$

So $r=2r+1$ or $r+2r+1=43$

$r=-1\Rightarrow$ not possible

So $3r+1=43$

$r=14$