Effective force constant is equal to the reciprocal of the sum of individual force constant, hence

$\frac{1}{k_e}=\frac{1}{k_1}+\frac{1}{k_2}+\frac{1}{k_3}+\dots$

Given, $k_1=k, k_2=2k, k_3=3k,\dots ..$

$\therefore$ $\frac{1}{k_e}=\frac{1}{k}+\frac{1}{2k}+\frac{1}{4k}+\frac{1}{8k}+\dots$

The given series a geometric progression series, hence sum is

$S_{\infty }=\frac{a}{1-r}$

where $a$ is first term of series and $r$ the common difference.

$\Rightarrow$ $\frac{1}{k_e}=\frac{1}{k}\times \frac{1}{\left(1-\frac{1}{2}\right)}=\frac{2}{k}\Rightarrow k_e=\frac{k}{2}$