Prove that `1/(n+1)+1/(n+2)+...+1/(2n)> 13/24` ,for all natural number

1.	Prove that `1/(n+1)+1/(n+2)+...+1/(2n)> 13/24` ,for all natural number `n>1`.
Answer» P(n) : `(1)/(n+1)+(1)/(n+2)+ . . .+(1)/(2n)gt(13)/(24)`, for all natural numbers `ngt1`. Step I We observe that, P(2) is true, `P(2):(1)/(2+1)+(1)/(2+2)gt(13)/(24).` `(1)/(3)+(1)/(4)gt(13)/(24)` `(4+3)/(12)gt(13)/(24)` `(7)/(12)gt(13)/(24)` which is true Step II Now, we assume that P(n) is true, For n=k, `P(k):(1)/(k+1)+(1)/(k+2)+ . . . +(1)/(2k)gt(13)/(24).` Step III Now, to prove P(k+1) true we have to show that `P(k+1):(1)/(k+1)+(1)/(k+2)+ . . . +(1)/(2k)+(1)/(2(k+1))gt(13)/(24)` Given. `(1)/(k+1)+(1)/(k+2)+ . . . +(1)/(2k)gt(13)/(24)` `(1)/(k+1)+(1)/(k+2)+ . . . +(1)/(2k)+(1)/(2(k+1))gt(13)/(24)+(1)/(2(k+1))` `(13)/(24)+(1)/(2(k+1))gt(13)/(24)` `because(1)/(k+1)+(1)/(k+2)+ . . . +(1)/(2k)+(1)/(2(k+1))gt(13)/(24)` So, P(k+1) is true, whenever p(k) is true. Hence, P(n) is true.

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