Question

Find the numerically greatest term in the expansion of $(7-5x{)}^{11}$ when $x=\frac{2}{3}$.

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

In the binomial expansion of $(x+y{)}^n$,
The greatest term occurs for r = $\left[\frac{(n+1)}{(1+|\frac{x}{y}|)}\right]$, where [] denotes the greatest integer function.

$\frac{(n+1)}{(1+|\frac{x}{y}|)}=\frac{(11+1)}{(1+|\frac{7}{5x}|)}$ (x on the LHS and RHS are different)

$=\frac{\left(12\right)}{(1+|\frac{7\times 3}{5\times 2}|)}$

$=\frac{\left(12\right)}{(1+|\frac{21}{10}|)}$

$=\frac{\left(12\right)}{\left(3.1\right)}$

$r=[\frac{\left(12\right)}{\left(3.1\right)}]$

=3

$\Rightarrow T_4$ is the numerically greatest term ($T_{r+1}$ is the greatest term, that's what we assume in the derivation)