Question

The values of $x$ for which the $4^{th}$ term in the expansion of $(5+3x{)}^{10}$ is the greatest are:

• A. $\frac{5}{8}\le x\le \frac{20}{21}$
• B. $\frac{5}{8}<x<\frac{20}{21}$
• C. $\frac{5}{7}<x<\frac{19}{21}$
• D. $-\frac{5}{8}<x<\frac{20}{21}$

Answer 1

The correct option is B $\frac{5}{8}<x<\frac{20}{21}$

We can write $(5+3x{)}^{10}$ as $5^{10}{(1+\frac{3x}{5})}^{10}$
The numerically greatest term is given by $\frac{(n+1)|x|}{1+|x|}$
It is given that the $4^{th}$ term is the largest.
Therefore $3<\frac{11\left(|\frac{3x}{5}|\right)}{1+|\frac{3x}{5}|}<4$
$3<\frac{|3x|11}{5+|3x|}<4$
Hence $15+9x<33x$
$15<24x$
$x>\frac{5}{8}$ and $20+12x>33x$
$20>21x$
$x<\frac{20}{21}$
Hence $x\in (\frac{5}{8},\frac{20}{21})$