In the expansion of (1 + a)m + n, prove that coefficients of amand anare equal - Maths - Binomial Theorem

Question

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

Gopal Mohanty · Accepted Answer

It is known that (r + 1)th term, (Tr+1), in the binomial expansion of (a + b)n is given by

Assuming that am occurs in the (r + 1)th term of the expansion (1 + a)m + n, we obtain

Comparing the indices of a in am and in Tr + 1, we obtain r = m Therefore, the coefficient of am is

Assuming that an occurs in the (k + 1)th term of the expansion (1 + a)m+n, we obtain

Comparing the indices of a in an and in Tk + 1, we obtain k = n Therefore, the coefficient of an is

Thus, from (1) and (2), it can be observed that the coefficients of am and an in the expansion of (1 + a)m + n are equal

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