Concept:

a + ar + ar2 + ar3 +….. 

Sum of the above infinite geometric series:

\(=\frac{a}{1-r}\)

Analysis:

Given:

1 + 2(a2 + 1) + 3(a2 + 1)2 + 4(a2 + 1)3 + ......

let x = (a2 + 1)

The series now becomes

S = 1 + 2x + 3x² + 4x³ + ......  ----(1)

By multiplying x on both sides we get

xS = x + 2x² + 3x³ + 4x⁴ + ...... ----(2)

Subtracting (1) and (2), we get

S(1 - x) = 1 + x + x² + x³ + ..... ---(3)

The right hand side of (3) forms infinite geometric series with a = 1, r = x

∴ S(1 - x) = \(\frac{1}{1-x}\)

\(\Rightarrow S = \frac{1}{(1-x)^2}\)

putting the value of x, we get

\(\Rightarrow S = \frac{1}{(1- a^2 - 1)^2}\)

\(\Rightarrow S = \frac{1}{a^4}\)