Question

Find the $4^{th}$ and $8^{th}$ term of the series $6,4,3,............$

Answer 1

Consider $1/6,1/4,1/3,............\infty$
Here $T_2-T_1=T_3-T_2=1/12\Rightarrow 1/6,1/4,1/3$ is an A.P.
$4^{th}$ term of this A.P. $=\frac{1}{6}+3\times \frac{1}{12}=\frac{1}{6}+\frac{1}{4}=\frac{5}{12}$,
And the $8^{th}$ term $=\frac{1}{6}+7\times \frac{1}{12}=\frac{9}{12}$.
Hence the $8^{th}$ term of the H.P. $=12/9=4/3$ and the $4^{th}term=12/5$.