What is the fourth term of the expansion of the binomial (2x + 5)⁵?

Solution:

Given, the expression is (2x + 5)⁵

We have to find the fourth term in the binomial expansion.

The general formula is \(t_{r+1}=\, ^{n}C_{r}\times a^{(n-r)}\times b^{r}\)

Here, a = 2x, b = 5, n = 5, r = 3

So, \(t_{3+1}=\, ^{5}C_{3}\times (2x)^{(5-3)}\times (5)^{3}\)

\(t_{4}=(\frac{5\times 4\times 3\times 2\times 1}{(3\times 2\times 1)(5-3)!})\times (2x)^{(2)}\times (5)^{3}\)

\(t_{4}=(\frac{5\times 4}{2\times 1})\times 4x^{2}\times 125\)

\(t_{4}=10\times 4x^{2}\times 125\)

\(t_{4}=5000x^{2}\)

Therefore, the fourth term is \(t_{4}=5000x^{2}\)

What is the fourth term of the expansion of the binomial (2x + 5)⁵?

Summary:

The fourth term in the binomial expansion of (2x + 5)⁵ is \(t_{4}=5000x^{2}\)