find the sum of n terms of the series (a+b) + (a2+2b) + (a3 + 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) + (a2 + 2b) + (a3 + 3b) + .....+ (an + nb) can be rewritten as
(a + a2 + a3 + ...... + an) + (b + 2b + 3b + ..... + nb)
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