find the sum of n terms of the series (a+b) + (a^{2}+2b) + (a^{3} + 3b = .....

The solution to the mentioned question requires a higher level of understanding (concept of G.P.) which is not taught in your grade still i am providing you the answer.

The given series i.e., (a + b) + (a^{2 }+ 2b) + (a^{3} + 3b) + .....+ (a^{n} + *n*b) can be rewritten as

(a + a^{2 }+ a^{3} + ...... + a^{n}) + (b + 2b + 3b + ..... + *n*b)

Clearly the first series forms a G.P. with first term a and common ratio also a.

and the second series forms an A.P. with first term b and common ratio also b.

Hence, required sum is

for a < 1

or

for a > 1

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