Question

For which positive values of x is the forth term in the expansion of ${(5+3x)}^{10}$
is the greatest

• A. $\frac{2}{21}<x<\frac{5}{8}$
• B. $\frac{5}{8}<x<\frac{20}{21}$
• C. $\frac{2}{21}<x<\frac{4}{5}$
• D. $\frac{4}{5}<x<\frac{20}{21}$

Answer 1

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

$T_4>T_5$ ...(i)
And $T_3<T_4$ ...(ii)
From i, we get
${}^{10}{C}_3{.5}^7.(3x{)}^3>{}^{10}{C}_4{5}^6.(3x{)}^4$
$\frac{{}^{10}{C}_3}{{}^{10}{C}_4}.\frac{5}{3}>x$
Or
$x<\frac{20}{21}$. ...(i)
From ii we get
${}^{10}{C}_2{5}^8(3x{)}^2<{}^{10}{C}_3{5}^7(3x{)}^3$
$\frac{{}^{10}{C}_2}{{}^{10}{C}_3}.\frac{5}{3}<x$
Or
$x>\frac{5}{8}$.
Hence
$\frac{5}{8}<x<\frac{20}{21}$.