In a $G.P.,$ the $3^{rd}$ term is $24$ and the $6^{\text {th }}$ term is $192 .$ Find the $10^{\text {th }}$ term.
Here, $a_{3}=a r^{2}=24$ ........$(1)$
and $a_{6}=a r^{5}=192$ ...........$(2)$
Dividing $(2)$ by $(1),$ we get $r=2 .$ Substituting $r=2$ in $(1),$ we get $a=6$
Hence $\quad a_{10}=6(2)^{9}=3072$
If $a,\;b,\;c$ are ${p^{th}},\;{q^{th}}$ and ${r^{th}}$ terms of a $G.P.$, then ${\left( {\frac{c}{b}} \right)^p}{\left( {\frac{b}{a}} \right)^r}{\left( {\frac{a}{c}} \right)^q}$ is equal to
If the sum of the series $1 + \frac{2}{x} + \frac{4}{{{x^2}}} + \frac{8}{{{x^3}}} + ....\infty $ is a finite number, then
Which term of the following sequences:
$\sqrt{3}, 3,3 \sqrt{3}, \ldots$ is $729 ?$
Let the positive numbers $a _1, a _2, a _3, a _4$ and $a _5$ be in a G.P. Let their mean and variance be $\frac{31}{10}$ and $\frac{ m }{ n }$ respectively, where $m$ and $n$ are co-prime. If the mean of their reciprocals is $\frac{31}{40}$ and $a_3+a_4+a_5=14$, then $m + n$ is equal to $.........$.
A person writes a letter to four of his friends. He asks each one of them to copy the letter and mail to four different persons with instruction that they move the chain similarly. Assuming that the chain is not broken and that it costs $50$ paise to mail one letter. Find the amount spent on the postage when $8^{\text {th }}$ set of letter is mailed.