sum of series (a+b) +(a^+2b)+(a^+3b)+....to n terms
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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
a (1-a^n)/(1-a)+n/2 [(n+1)b]
For a > 1
a (a^n-1)/(a-1)+n/2[(n+1)6]
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
a (1-a^n)/(1-a)+n/2 [(n+1)b]
For a > 1
a (a^n-1)/(a-1)+n/2[(n+1)6]
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