Mathematics

Mathematics

Problem

Let nn be a positive integer. For the binomial expansion of (1+x)n(1+x)^n answer the following questions. a. (i) Define what is meant by “the sum of the coefficients” in the expansion of (1+x)n(1+x)^n. b. (ii) Show, without actually expanding (1+x)n(1+x)^n, that the sum of the coefficients is 2n2^n. c. (iii) For n=8n=8, find separately the sum of the coefficients of the terms that contain an even power of xx and the sum of the coefficients of the terms that contain an odd power of xx. Give reasons for each answer.
Intuitive explanation

The binomial expansion (1+x)n(1+x)^n generates coefficients that follow a beautiful pattern—the binomial coefficients (nk)\binom{n}{k}. The key insight is that each coefficient tells you how many ways to choose kk items from nn, which connects algebra to combinatorics. Understanding this link lets you extract coefficients, find sums, and solve counting problems using the expansion itself.

Intuitive explanation

The binomial expansion (1+x)n(1+x)^n generates coefficients that follow a beautiful pattern—the binomial coefficients (nk)\binom{n}{k}. The key insight is that each coefficient tells you how many ways to choose kk items from nn, which connects algebra to combinatorics. Understanding this link lets you extract coefficients, find sums, and solve counting problems using the expansion itself.

Browse the problem library