Question bank/JEE Main/Mathematics

A game is played with a special fair cubic die which has one red
side, two blue sides, and three green sides. The result is the colour of
the top side after the die has been rolled. If the die is rolled
repeatedly the probability that the second blue result occurs on or
before the tenth roll, can be expressed in the form
$\frac{3^{p} - 2^{q}}{3^{r}}$ where p, q, r are positive integers, then
the value of $p^{2} + q^{2} + r^{2}$ is

A243B283C226D209

Show answer and solution▾

$p = \left( \frac{1}{3} \right)^{2}\left\{ 1 + 2\left( \frac{2}{3} \right) + 3\left( \frac{2}{3} \right)^{2} + \ldots + 9\left( \frac{2}{3} \right)^{8} \right\}$
$s = 1 + 2\left( \frac{2}{3} \right) + \ldots.9\left( \frac{2}{3} \right)^{8}$
$\frac{2}{3}s = \left( \frac{2}{3} \right) + \ldots + 9\left( \frac{2}{3} \right)^{9}$
$\frac{s}{3} = 1 + \frac{2}{3}\ldots\left( \frac{2}{3} \right)^{8} - 9\left( \frac{2}{3} \right)^{9}$
$s = 9 - \left( \frac{2}{3} \right)^{9}36$
$p = 1 - \left( \frac{2}{3} \right)^{9} \cdot 4 = \frac{3^{9} - 2^{11}}{3^{9}}$

Source: Grass-G / M-23 Mathematics Paper 31 Dec FST.docx