Find the sum of this seemingly random sequence!

Relevant wiki: Riemann Zeta Function

$\begin{aligned} S & = \frac {1.5}5 + \frac {1.5}{20} + \frac 1{30} + \frac {1.5}{80} + \frac {1.5}{125} + \frac 1{120} + \cdots \\ & = \frac 3{10} + \frac 3{40} + \frac 3{90} + \frac 3{160} + \frac 3{250} + \frac 3{360} + \cdots \\ & = \frac 3{10}\sum_{k=1}^\infty \frac 1{k^2} \\ & = \frac 3{10}\color{#3D99F6} \zeta (2) & \small \color{#3D99F6} \text{where }\zeta (\cdot) \text{ denotes the Riemann zeta function.} \\ & = \frac 3{10}\times \color{#3D99F6} \frac {\pi^2}6 \\ & = \frac {\pi^2}{20} \approx \boxed{0.493} \end{aligned}$

Sum of the sequence = 1.5/5 + 1.5/20 + 1/30 + 1.5/80 + 1.5/125 + 1/120 + ..........

                                    =   3/10 + 3/40 + 3/90 + 3/160 + 3/250 + 3/360 .........

                                    =   3/10 * (1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + ..........  )

                                    =   3/10*(pi^2 / 6)

, which is approximately equal to 0.49348022005