It is known that (r + 1)th term, (Tr+1), in the binomial expansion of (a + b)n is given by
Assuming that xn occurs in the (r + 1)th term of the expansion of (1 + x)2n, we obtain
Assuming that xn occurs in the (k +1)th term of the expansion (1 + x)2n – 1, we obtain
Therefore,
the coefficient of xn in the expansion of (1 + x)2n is twice the coefficient of xn in the expansion of (1 + x)2n–1.
Hence, proved.