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

Explanation

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.