Problem 2 - Beginning and Intermediate Algebra

Question

What are the first and last terms in the expansion of \((a+b)^{n} ?\)

Step-by-Step Solution

Answer

The first term in the expansion of \((a+b)^{n}\) is \(a^n\), and the last term is \(b^n\).

Step 1: Calculate the first term in the expansion

To calculate the first term in the expansion, substitute k = 0 into the binomial theorem formula: \((a+b)^{n} = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^{k}\) First term = \(\binom{n}{0} a^{n-0} b^{0}\) Now, we know that \(\binom{n}{0} = 1\) and \(b^0 = 1\). Therefore, the first term in the expansion is: First term = \(a^n\)

Step 2: Calculate the last term in the expansion

Similarly, to find the last term in the expansion, substitute k = n into the binomial theorem formula: \((a+b)^{n} = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^{k}\) Last term = \(\binom{n}{n} a^{n-n} b^{n}\) We also know that \(\binom{n}{n} = 1\), \(a^0 = 1\) and \(b^n\) is the power of b raised to n. Hence, the last term in the expansion is: Last term = \(b^n\) Now, we have determined the first and last terms in the expansion of \((a+b)^{n}\). First term: \(a^n\) Last term: \(b^n\)