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Question

Let Sn=113+1+213+23+1+2+313+23+33+......+1+2+.....+n13+23+....+n3. If 100Sn=n, then n is equal to :

A
199
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B
99
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C
200
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D
19
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Solution

The correct option is A 199
Sn=113+1+213+23+1+2+313+23+33+.....+1+2+......+n13+23+......+n3

Tr=rk=1krk=1k3=r(r+1)2[r(r+1)2]2=2r(r+1)=2[r+1r]r(r+1)=2r2r+1

Sn=nr=12r2r+1

=2122+2223+2324+.......+2n2n+1

=22n+1

100Sn=n100(22n+1)=n

100(2n+22)=n(n+1)

200n=n(n+1)

n2199n=0

n(n199)=0

n=0 or n=199

n cannot be 0

Hence, n=199.

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