Chapter SH, Problem 10.22EP

To determine

To find:The third term of the given expression.

Answer to Problem 10.22EP

The third term is $240a^2{b}^4$ .

Explanation of Solution

Given: The expression is ${\left(a+2b\right)}^6$

Consider the expression,

  ${\left(a+2b\right)}^6$ ...... (1)

The expansion of the binomial expansion ${\left(a+b\right)}^n$ is given by,

  ${\left(a+b\right)}^n={\displaystyle \sum _{k=0}^n\left(\begin{array}{l}n\\ k\end{array}\right)a^{n-k}b}$

So,

  ${\left(a+2b\right)}^6={\displaystyle \sum _{k=0}^6\left(\begin{array}{l}6\\ k\end{array}\right){\left(2b\right)}^{6-k}k}$

The third term shall be at $k=2$ .

  $\begin{array}{c}{\left({\left(a+2b\right)}^6\right)}_{\text{3rd}\text{term}}=\frac{6!}{\left(6-2!\right)2!}{\left(2b\right)}^4{\left(a\right)}^2\\ =240a^2{b}^4\end{array}$

Therefore, the third term is $240a^2{b}^4$ .