We have to find average of the coefficients of the two middle terms in the expansion of (1+x)2n+1 Since, 2n + 1 is odd. Hence (
2n+1+1
2
) and (
2n+1+3
2
) are two middle ⇒(n+1)th and (n+2)th term are two middle terms. Now, coefficients of the (n+1)th term and (n+2)th term are 2n+1cn and 2n+1cn+1 Average of the coefficients of the two middle terms =
2n+1cn+2n+1cn+1
2
We know that ncr+ncr+1=n+1cr+1 So, 2n+1cn+2n+1cn+1=2n+2cn+1 Average of the coefficients of the two middle terms =
2n+1cn+2n+1cn+1
2
=
2n+2cn+1
2
We know that ncr=
n
r
n−1cr−1 So, 2n+2cn+1=
2n+22n+1
n+1
cn=
2(n+1)
n+12n+1
cn=2×2n+1cn Average of the coefficients of the two middle terms =