Question

Prove that an integer is odd if and only if it is the sum of two consecutive integers.

Answer 1

A proof can be as follows:

Explanation:

Remember that an odd interger is of the form
`2p+1` where
`p` is a integer and remember that two consecutive integer are two numbers of the form
`p, p+1`

`(\Rightarrow)` Suppose the
`n` is an odd integer.

Then
`n-1` must be an even integer and hence divisible by 2. Then we define

`p=(n-1)/(2)\\q=(n-1)/(2)+1`

Then we have that

`p+q=(n-1)/(2)+(n-1)/(2)+1=((n-1)+(n-1))/(2)+1=(2(n-1))/(2)+1=n-1+1=n`

The converse is as follows:

`(\Leftarrow)` Let
`p` an integer, then
`p,p+1` are two consecutive integers. Then

`n=p+(p+1)=2p+1` is an odd integer.