In the expansion of (1 + a)^m+n, prove that coefficients of a^m and aⁿ are equal.

The general term T_r+1 in the binomial expansion is given by T_r+1 = ⁿC_r a^n-r b^r

Here n= m+n , a = 1 and b= a

Putting the values in the general form

T_r+1 = ^m+nC_r 1^m+n-r a^r

= ^m+nC_r a^r………….1

The coefficient of a^m is & the coefficient of aⁿ is

a^r = a^m aⁿ = a^r

⇒ r=m ⇒ n =r

Putting r = m in 1 & putting r= n in 1

T_m+1= ^m+nC_m1^m+n-m a^mT_n+1 = ^m+nC_n1^m+n-n aⁿ

&

&

The coefficient of a^m and aⁿ are same i.e.;

Hence proved.