Chapter 11, Problem 22CLR

To determine

To calculate:

The twelfth term in the following expansion ${\left(2x+y\right)}^{13}$

Answer to Problem 22CLR

The twelfth term value is $312x^2{y}^{11}$ .

Explanation of Solution

Given information:

  ${\left(2x+y\right)}^{13}$

Formula used:

  $t_{r+1}=C{}_r\times a^{n-r}\times b^r$

Calculation:

Consider the expansion:

  ${\left(2x+y\right)}^{13}$

By using this formula:

  $\begin{array}{l}{t}_{r+1}=C{}_r\times a^{n-r}\times b^r\\ t_{12}=C{}_{11}\times 2x^{13-11}\times y^{11}\mathrm{...}\left(1\right)\end{array}$

The solution of $C{}_{11}$ is

  $\begin{array}{l}C{}_r=C\left(13,11\right)\\ C{}_r=\frac{13!}{\left(11!\left(13-11\right)!\right)}\\ C{}_r=78\end{array}$

Put this value in equation (1)

  $\begin{array}{l}{t}_{12}=C{}_{11}\times 2x^2\times y^{11}\\ t_{12}=78\times 4x^2\times y^{11}\\ t_{12}=312x^2{y}^{11}\end{array}$

The twelfth term value is $312x^2{y}^{11}$ .