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

Question:

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.

Solution:

Coefficient of the $(r+1)$ th term in $(1+x)^{n+1}$ is ${ }^{n+1} C_{r}$

Sum of the coefficients of the $r$ th and $(r+1)$ th terms in $(1+x)^{n}={ }^{n} C_{r-1}+{ }^{n} C_{r}$

$={ }^{n+1} C_{r} \quad\left[\because{ }^{n} C_{r+1}+{ }^{n} C_{r}={ }^{n+1} C_{r+1}\right]$

Hence proved.

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