Question

The square of the sum of two consecutive integers is 1681. What are the integers?

Answer 1

20 and 21.

Explanation:

Let's say the two consecutive numbers are `a` and `b`. We need to find an equation that we can solve to work out their values.

"The square of the sum of two consecutive integers is `1681`." That means if you add `a` and `b` together, then square the result, you get `1681`. As an equation we write:

`(a+b)^2=1681`

Now, there are two variables here so at first glance it looks unsolvable. But we're also told that `a` and `b` are consecutive, which means that `b=a+1`!

Substituting this new information in gives us:

`(a+a+1)^2=1681`
`(2a+1)^2=1681`

Next we're going to follow these steps to solve for `a`:

1) Take the square root of both sides. This will give two possible results, since both positive and negative numbers have positive squares.
2) Subtract `1` from both sides.
3) Divide both sides by `2`.
4) Check the answer.

`(2a+1)^2=1681`
`2a+1=sqrt(1681)=41`
`2a=40`
`a=20`

This means that `b=21`! To check these answers, take the values `20` and `21` and substitute them into the original equation like this:

`(a+b)^2=1681`
`(20+21)^2=1681`
`1681=1681`

Success!