Question

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

Answer 1

The given G.P. is $\frac{5}{2},\frac{5}{4}\frac{5}{8},...$
Here, $a=$ first term $=\frac{5}{2},$ $r=$ common ratio $=\frac{\frac{5}{4}}{\frac{5}{2}}=\frac{1}{2}$
$\therefore a_{20}={ar}^{20-1}=\frac{5}{2}{\left(\frac{1}{2}\right)}^{19}=\frac{5}{\left(2\right){\left(2\right)}^{19}}=\frac{5}{{\left(2\right)}^{20}}$
and $a_n={ar}^{n-1}=\frac{5}{2}{\left(\frac{1}{2}\right)}^{n-1}=\frac{5}{\left(2\right){\left(2\right)}^{n-1}}=\frac{5}{{\left(2\right)}^n}$