Problem 61 - College Algebra

Question

Describe the pattern in the exponents on \(b\) in the expansion of \((a+b)^{n}\).

Step-by-Step Solution

Answer

In the expansion of \((a+b)^{n}\), the exponents on \(b\) start at 0 and increase by 1 with each term, until reaching \(n\) in the final term.

Step 1: Understanding the Binomial Theorem

The binomial theorem states that \((a+b)^n = \sum_{k=0}^{n} {n\choose k} a^{n-k} b^{k}\), where \({n\choose k}\) represents the number of ways to choose \(k\) items from a set of \(n\) (also called binomial coefficients), \(a^{n-k}\) represents the term involving the variable \(a\) and \(b^{k}\) represents the term involving the variable \(b\).

Step 2: Identifying the pattern of exponents of \(b\)

From the Binomial Theorem, it can be observed that the exponent of \(b\) starts from 0 and increases by 1 in each term till it reaches \(n\) in the final term of the expansion.