The sum of the series 1+2x+3x2+4x3+..... upto n terms is
A
1−(n+1)xn+nxn+1(1−x)2
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B
1−xn(1−x
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C
xn+1
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D
None of these
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Solution
The correct option is A1−(n+1)xn+nxn+1(1−x)2 Let Sn be the sum of the given series to n terms, then Sn=1+2x+3x2+4x3+....+nxx−1....(i) xSn=x+2x2+3x2+....+nxn....(ii)
Subtracting (ii) from (i), we get (1−x)Sn=1+x+x2+x3+.... to n terms −nxn =((1−xn)(1−x)−nxn) ⇒Sn=(1−xn)−nxn(1−x)(1−x)2=1−(n+1)xn+nxn+1(1−x)2.