Question

question
find the 5th term in the expansion of ((a + b)^5) in simplest form.

Explanation:

Step1: Recall the binomial theorem

The binomial theorem states that \((a + b)^n=\sum_{k = 0}^{n}\binom{n}{k}a^{n - k}b^{k}\), where \(\binom{n}{k}=\frac{n!}{k!(n - k)!}\) and the \((k + 1)\)-th term is given by \(\binom{n}{k}a^{n - k}b^{k}\).

Step2: Determine the value of \(k\) for the 5th term

For the 5th term, we know that the term number is \(k+1 = 5\), so \(k=4\). Here, \(n = 5\).

Step3: Calculate the binomial coefficient \(\binom{5}{4}\)

Using the formula \(\binom{n}{k}=\frac{n!}{k!(n - k)!}\), we substitute \(n = 5\) and \(k = 4\):
\[
\binom{5}{4}=\frac{5!}{4!(5 - 4)!}=\frac{5!}{4!1!}=\frac{5\times4!}{4!\times1}=5
\]

Step4: Find the powers of \(a\) and \(b\)

For the \(k = 4\) term, \(a\) has the power \(n-k=5 - 4=1\) and \(b\) has the power \(k = 4\).

Step5: Form the 5th term

Multiply the binomial coefficient, \(a^{n - k}\), and \(b^{k}\) together:
\[
\binom{5}{4}a^{5 - 4}b^{4}=5\times a^{1}\times b^{4}=5ab^{4}
\]

Answer:

\(5ab^{4}\)