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Question

# The sum of the series 1 + 2x + 3 x2 + 4 x3 + ..........upto n terms is

A

1(n+1)xn+nxn+1(1x)2

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B

1xn1x

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C

xn+1

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D

None of these

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Solution

## The correct option is A 1−(n+1)xn+nxn+1(1−x)2 Let Sn be the sum of the given series to n terms, then Sn = 1 + 2x + 3 x2 + 4 x3 + ....... + n xn−1 ..........(i) xSn = x + 2 x2 + 3 x2 + ...........+ n xn ..........(ii) Subtracting (ii) from (i), we get (1-x)Sn = 1 + x + x2 + x3 + ......... to n terms -n xn = ( (1−xn)(1−x)) - nxn ⇒ Sn = (1−xn)−nxn(1−x)(1−x)2 = 1−(n+1)xn+nxn+1(1−x)2

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