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.

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Solution

$Coefficient of the (r + 1) th term in (1 + x)^{n + 1} is {}^{n + 1}C_{r} Sum of the coefficients of the r th and (r + 1) th terms in (1 + x)^{n} = {}^{n}C_{r - 1} + {}^{n}C_{r} = {}^{n + 1}C_{r} [∵ {}^{n}C_{r + 1} + {}^{n}C_{r} = {}^{n + 1}C_{r + 1}] Hence proved .$

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