Thursday, June 29, 2017

summation - Find the sum of the n terms of the series 2cdot20+3cdot21+4cdot22+dots



Find the sum of the n terms of the series:





220+321+422+




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

nk=02kxk+2=x22n+1xn+312x
Differentiating (1) yields
nk=0(k+2)2kxk+1=2x(n+3)2n+1xn+212x+2x22n+2xn+3(12x)2
Plugging in x=1 leads to
nk=0(k+2)2k=(n+1)2n+1


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