Question

How do you find all sets of three consecutive odd integers whose sum is between 20 and 30?

Answer 1

Write the odd numbers in the form `2k+1, 2k+3, 2k+5` and set up an inequality to find all valid values for `k` to find that the sets are
`{5, 7, 9}` and `{7, 9, 11}`

Explanation:

Any set of three consecutive odd integers may be written as
`{2k+1, 2k+3, 2k+5}` for some `k in ZZ`

Then, we just need to find a condition on `k` such that
`20<(2k+1)+(2k+3)+(2k+5)<30`

Simplifying, we get

`20<6k+9<30`

Subtracting `9` gives us

`11 < 6k < 21`

Dividing by `6`

`11/6 < k < 21/6`

Thus we will have a set with the desired property when `k` is an integer between `11/6` and `21/6`, that is, when `k=2` or `k=3`. This gives us the result that the only such sets are

`{2(2)+1, 2(2)+3, 2(2)+5} = {5, 7, 9}`
and
`{2(3)+1, 2(3)+3, 2(3)+5} = {7, 9, 11}`