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Question

For any natural number n > 1
1n+1+1n+2+.....+12n>1314

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Solution

Let Sn=1n+1+1n+2+....+12
For n=2, we have
13+14=712=1424>1324
Hence the inequality holds for n=2
Now assume Sm>1324 for some positive integer m>1
We have Sm=1m+1+1m+2+....+12m
and Sm+1=1m+2+1m+3+....+12m+12m+1+12m+2
Sm+1Sm=12m+1+12m+21m+1
that is , Sm+1Sm=12(m+1)(2m+1)>0 for any natural number m . It follows that Sm+1Sm>0,thatisSm+1>Sm
But Sm>1324 by our assumption
Sm+1>1324
Hence by mathematical induction . The given inequality holds for all natural numbers n >1

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