Prove that the coefficient of (r + 1)th term in the expansion of (1 +

1.	Prove that the coefficient of (r + 1)th term in the expansion of (1 + x)n + 1 is equal to the sum of the coefficients of rth and (r + 1)th terms in the expansion of (1 + x)n.
Answer» As we know, the coefficients of (r + 1)^th term in (1 + x)ⁿ⁺¹ is ⁿ⁺¹C_r Therefore, sum of the coefficients of the r^th and (r + 1)^thterms in (1 + x)ⁿ is (1 + x)ⁿ = ⁿC_r-1 + ⁿC_r = ⁿ⁺¹C_r [since, ⁿC_r+1 + ⁿC_r = ⁿ⁺¹C_r+1] Thus proved.

Prove that the coefficient of (r + 1)th term in the expansion of (1 + x)n + 1 is equal to the sum of the coefficients of rth and (r + 1)th terms in the expansion of (1 + x)n.

Answer»

As we know, the coefficients of (r + 1)^th term in (1 + x)ⁿ⁺¹ is ⁿ⁺¹C_r

Therefore, sum of the coefficients of the r^th and (r + 1)^thterms in (1 + x)ⁿ is

(1 + x)ⁿ = ⁿC_r-1 + ⁿC_r

= ⁿ⁺¹C_r [since, ⁿC_r+1 + ⁿC_r = ⁿ⁺¹C_r+1]

Thus proved.