Question

If $\left(16\right)^{15} +2\cdot17\left(16\right)^{14} +3\left(17\right)^{2} \left(16\right)^{13} + ....+16\left(17\right)^{15} =m \left(16\right)^{15}$ then value of $\sqrt{m +33}$ equals

• A. $15$
• B. $16$
• C. $17$
• D. $18$

Answer 1

Answer: (C)

$\because m\left(16\right)^{15} =\left(16\right)^{15} +2 \cdot17\left(16\right)^{14} +3\left(17\right)^{2} \left(16\right)^{13} +....+16^{1}\left(17\right)^{15}$
$\therefore m =1 +\frac{2 \cdot17\left(16\right)^{14}}{\left(16\right)^{15}} + 3\cdot\frac{\left(17\right)^{2}\left(16\right)^{13}}{\left(16\right)^{15}} + ...+\frac{16\left(17\right)^{15}}{\left(16\right)^{15}}$
or $m = 1 +2\left(\frac{17}{16}\right) +3\left(\frac{17}{16}\right)^{2} + ...+16\left(\frac{17}{16}\right)^{15}$
or $m = 1 +2x +3x^{2} +...+16x^{15}$ (where $x=\frac{17}{16}$)
$= \frac{d}{dx} \left(x^{1} +x^{2} +....+x^{16}\right)$
$= \frac{d}{dx} \left[\frac{x \left(x^{16} -1\right)}{x -1}\right]$
$= \frac{\left(x -1\right)\left(17 x^{16}-1\right) -x\left(x^{16} -1\right)}{\left(x -1\right)^{2}} = \frac{16x^{17} -17x^{16} +1}{\left(x -1\right)^{2}}$
$\Rightarrow m = \frac{1}{\left(x -1\right)^{2}}$ ($\because16x^{17} =17x^{16}$ where $x = \frac{17}{16}$)
$\Rightarrow m = 256$
$\therefore m +33 =289 \Rightarrow \sqrt{m +33} =17$