Question

Find the ${20}^{th}$ and $n^{th}$ terms of the G.P. $\frac{5}{2},\frac{5}{4},\frac{5}{8},...$

Answer 1

Given G.P. $\frac{5}{2},\frac{5}{4},\frac{5}{8},...$
Here, First term $=a=\frac{5}{2}$ and
common ratio $=r=\frac{\frac{5}{4}}{\frac{5}{2}}=\frac{5}{4}\times \frac{2}{5}=\frac{1}{2}$

Step $1:$ Finding $n^{th}$ term of G.P.
we know the $n^{th}$ term of G.P. is
$a_n=a{r}^{n-1}$
$\Rightarrow a_n=\frac{5}{2}{\left(\frac{1}{2}\right)}^{n-1}$
$\Rightarrow a_n=\frac{5}{2\cdot 2^{n-1}}$
$\Rightarrow a_n=\frac{5}{(2{)}^{1+n-1}}$
$\therefore a_n=\frac{5}{2^n}\cdots \left(i\right)$

Step $2:$ Finding ${20}^{th}$ term of G.P.
For ${20}^{th}$ term of G.P., putting $n=20\cdots \left(i\right)$
we have $a_{20}=\frac{5}{2^{20}}$

Hence, $n^{th}$ term of G.P. $=\frac{5}{2^n}$
and ${20}^{th}$ term of $G.P.=\frac{5}{2^{20}}$