The number of ways of selecting 3 numbers that are in G.P. from the set $\{1,2,3, \ldots, 100\}$ is
The number of ways of selecting 3 numbers that are in G.P. from the set $\{1,2,3, \ldots, 100\}$ is
18
52
14
53
Solution
Let three numbers in G.P. be $a$, $a r, a r^2$, such that
$a r^2 \leq 100 \Rightarrow a \leq \frac{100}{r^2}$ and $a$ be a positive integer.
So, $a=\left[\frac{100}{r^2}\right], 2 \leq r \leq 10$
$\begin{aligned}
&=\left[\frac{100}{4}\right]+\left[\frac{100}{9}\right]+\left[\frac{100}{16}\right]+\left[\frac{100}{25}\right]+\left[\frac{100}{36}\right] \\
&+\left[\frac{100}{49}\right]+\left[\frac{100}{64}\right]+\left[\frac{100}{81}\right]+\left[\frac{100}{100}\right]
\end{aligned}$
$=25+11+6+4+2+2+1+1+1=53 .$