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Question

If α be the nth root of unity then the sum of the series
1+2α+3α2+...nαn1 equals.

A
n1α
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B
n1α2
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C
n1α
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D
None of these
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Solution

The correct option is B n1α
S=1+2α+3α2+...nαn1
Here S forms an A.G.P
Therefore
S=1+2α+3α2+...nαn1
S(α)=α+2α2+3α3+...(n1)αn1+nαn
Hence
S(1α)=1+α+α2+....αn1nαn
S(1α)=1αn1αnαn
S(1α)=0n
S=n1α
Hence, option 'A' is correct.

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