Find the sum of the n terms of the series:
2⋅20+3⋅21+4⋅22+…
I don't know how to proceed. Please explain the process and comment on technique to solve questions of similar type.
Source: Barnard and Child Higher Algebra.
Thanks in Advance!
Answer
The formula for the sum of a geometric sequence gives
n∑k=02kxk+2=x2−2n+1xn+31−2x
Differentiating (1) yields
n∑k=0(k+2)2kxk+1=2x−(n+3)2n+1xn+21−2x+2x2−2n+2xn+3(1−2x)2
Plugging in x=1 leads to
n∑k=0(k+2)2k=(n+1)2n+1
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