Question

The sum of three consecutive odd integers is `-1317`. What is the smallest number?

Answer 1

`-441`

Explanation:

An odd number can be represented as `2n+1`. Since we want consecutive odds we will use `2n+1`, `2n+3`, and `2n+5`. We want the sum to be `-1317`, so:

`(2n+1)+(2n+3)+(2n+5)=-1317`

collect like terms:

`6n +9 =-1317`

subtract 9 from both sides:

`6n = -1326`

divide through by 6:

`n = -221`

Since `n=-221` our three numbers are:

`2(-221)+1 = -441`
`2(-221)+3 = -439`
`2(-221)+5= -437`

Of these the smallest is `-441` since it is less than the other numbers.

Answer 2

`-441`

Explanation:

We can represent any generic odd integer by the term:

`2n+1`

Hence, we can represent three consecutive odd integers by using `n, n+1, n+2` in the above expression, forming the sum:

`{2(n)+1} + {2(n+1)+1} + {2(n+2)+1} = -1317`

Expanding we have:

`(2n+1) + (2n+2+1) + (2n+4+1) = -1317`

Then collecting terms and solving for `n` we have:

`6n + 9 = -1317`
`:. 6n = -1326`
`:. n = -221`

With this value of `n` the three consecutive terms with sum `-1317` are:

`2(-221)+1 = -441`
`2(-221+1)+1 = -439`
`2(-221+2)+1 = -437`

Hence, the smallest of these numbers is `-441`