Question

Find the indicated term of each expansion.

Fifth term of ${\left(2a-b\right)}^7$

Short answer

The fifth term of the expansion is$T_5=280a^3{b}^4$.

Step-by-step solution

Step 1. Given Information.

Given, binomial expansion is ${\left(2a-b\right)}^7$ and the fifth term of the expansion is to be determined.

Step 2. Calculation.

The general term of the binomial expansion ${\left(x+y\right)}^n$ is given by:

$T_{r+1}=\frac{n!}{\left(n-r\right)!r!}{x}^{n-r}{y}^r$

Here n=7, x = 2a, y = -b and r + 1 = 5.

Plugging the values:

$\begin{array}{l}{T}_{r+1}=\frac{n!}{\left(n-r\right)!r!}{x}^{n-r}{y}^r\\ T_5=\frac{7!}{\left(7-4\right)!4!}{\left(2a\right)}^{7-4}{\left(-b\right)}^4\\ T_5=\frac{7!}{3!4!}{\left(2a\right)}^3{b}^4\\ T_5=280a^3{b}^4\end{array}$

Step 3. Conclusion.

Hence, the third term of the expansion is $T_5=280a^3{b}^4$.