(x + 2) is raised to the fifth power. the third term of the expansion is:

Solution:

Given, the expression is (x + 2)⁵

We have to find the third term of the expansion.

Binomial expansion of an exponential terms states that

\((x+y)^{n}=\sum_{r=0}^{n}\, ^{n}C_{r}x^{(n-r)}y^{r}\)

Where, \(^{n}C_{r}=\frac{n!}{r!(n-r)!}\)

Here, x = x, y = 2, n = 5

\(\\(x+2)^{5}=\, ^{5}C_{0}x^{(5-0)}(2)^{0}+\, ^{5}C_{1}x^{(5-1)}(2)^{1}+\, ^{5}C_{2}x^{(5-2)}(2)^{2}+\, ^{5}C_{3}x^{(5-3)}(2)^{3}+.....\\=\, ^{5}C_{0}x^{5}(1)+\, ^{5}C_{1}x^{4}(2)+\, ^{5}C_{2}x^{3}(4)+\, ^{5}C_{3}x^{2}(8)+.....\\=\, ^{5}C_{0}x^{5}+2\, ^{5}C_{1}x^{4}+4\, ^{5}C_{2}x^{3}+8\, ^{5}C_{3}x^{2}+.....\)

Therefore, the third term is \(4\, ^{5}C_{2}x^{3}\).

(x + 2) is raised to the fifth power. the third term of the expansion is:

Summary:

(x + 2) is raised to the fifth power. The third term of the expansion is \(4\, ^{5}C_{2}x^{3}\).