Problem 60 - College Algebra

Question

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

Step-by-Step Solution

Answer

In the expansion of \((a+b)^{n}\), the exponents on \(a\) start at \(n\) and decrease by one for each subsequent term until it reaches 0.

Step 1: Recognize the binomial theorem

The binomial theorem states that \((a+b)^n = \sum_{k=0}^{n} {n\choose k} a^{n-k} * b^k\). This means the sum of the terms will always equal \(n\) in each term of the expansion.

Step 2: Analyze the binomial coefficients and terms

Observe that in each term of the expansion, the exponent of \(a\) starts from \(n\) (the exponent in the original binomial) and decreases by 1 until it reaches 0. And the exponent of \(b\) starts from 0 and increases by 1 until it reaches \(n\). This is where the pattern lies in.

Step 3: Describe the pattern

So, in the expansion, the exponents on \(a\) start at \(n\) and decrease by one for each subsequent term until reaching 0. So, the pattern in the exponents on \(a\) in the expansion of \((a+b)^{n}\) is that they decrease from \(n\) to 0 as we move through each term of the expansion.