Problem 60 - College Algebra

Question

Explain how to find a particular term in a binomial expansion without having to write out the entire expansion.

Step-by-Step Solution

Answer

The \(r^{th}\) term in a binomial expansion \( (a+b)^n \) can be calculated directly using the formula \( ^nC_{r-1} * a^{n-r+1} * b^{r-1} \), obviating the need to unwrap the entire expansion. Here, \( ^nC_{r-1} \) denotes the binomial coefficient.

Step 1: Identify the necessary parameters

First, identify the values of \(a\), \(b\), \(n\) and \(r\) from your binomial. \( (a+b)^n \) is your binomial where \(a\) and \(b\) are the binomial terms and \(n\) is the power. \(r\) is the term number that you want to find.

Step 2: Apply the binomial theorem

Next, apply these values to the binomial theorem. The binomial theorem stipulates that the \(r^{th}\) term of the expansion can be evaluated using the formula \( ^nC_{r-1} * a^{n-r+1} * b^{r-1} \).

Step 3: Compute the binomial coefficient

Compute \( ^nC_{r-1} \), where \(C\) stands for combination. Use the combination formula \( ^nC_{r-1}= \frac{n!}{(n-r+1)!(r-1)!} \), where \(n!\) denotes the factorial of \(n\), that means, \(n*(n-1)*(n-2)*…..*3*2*1\). Calculate this value.

Step 4: Calculate a to the power and b to the power

Then, calculate the powers of \(a\) and \(b\) i.e. \(a^{n-r+1}\) and \(b^{r-1}\).

Step 5: Multiply the results

Finally, multiply the results obtained from the previous steps i.e. multiply the value obtained from the combination with the values obtained from the powers of \(a\) and \(b\) to get the required \(r^{th}\) term.