Question

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

Answer 1

Given : $(1+x{)}^{43}$

We know that the coefficient of the rth term in the expansion of $(1+x{)}^n$ is ${}^n{C}_{r-1}$

Therefore, the coefficient of the (2r+1)th and (r+2)th terms in the given expression are ${}^{43}{C}_{2r+1-1}$ and ${}^{43}{C}_{2r+1-1}$ and ${}^{43}{C}_{r+2-1}$ For these coefficient to be equal, we must have:

$\Rightarrow 2r=r+10$ or 2r+r+1 =43

$[\because {}^n{C}_r={}^n{C}_s\Rightarrow r=s$ or r+s =n]

$\Rightarrow r=14$

$[\because$ for r =1 it gives the same term]