Question

The question is below?

Show that `1/(n+1)+1/(n+2)+....+1/(2n)>=1/2`

Answer 1

Below

Explanation:

This can be proved by using mathematical induction

Step 1: Prove true for n=1

LHS: `1/(2times1)=1/2`
RHS: `1/2`

Since LHS=RHS, then it is true for n=1

Step 2: Assume true for n=k where k is a positive integer, `k>=1`
`1/(k+1)+1/(k+2)+...+1/(2k) >=1/2` --- (1)

Step 3: When `n=k+1`,
RTP: `1/(k+1)+1/(k+2)+...+1/(2k)+1/(2(k+1)) >=1/2`
ie `1/(k+1)+1/(k+2)+...+1/(2k)+1/(2(k+1))-1/2 >=0`

LHS:
`1/(k+1)+1/(k+2)+...+1/(2k)+1/(2(k+1))-1/2`

`<1/2+1/(2(k+1))-1/2` from (1) by assumption

`=1/(2(k+1))`

`=1/(2(k+1)) >0` (since k is an integer where `k>=1`)

`=RHS`

Therefore, true for `n=k+1`

Step 4: By proof of mathematical induction, it is true for all integers n greater than or equal to 1.

(This depends on what your school tells you to write so its better to ask your teacher what they want you to write)