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

Jan 8, 2016

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:

${\left(a + b\right)}^{2} = 1681$

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:

${\left(a + a + 1\right)}^{2} = 1681$
${\left(2 a + 1\right)}^{2} = 1681$

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 $2$.

${\left(2 a + 1\right)}^{2} = 1681$
$2 a + 1 = \sqrt{1681} = 41$
$2 a = 40$
$a = 20$

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

${\left(a + b\right)}^{2} = 1681$
${\left(20 + 21\right)}^{2} = 1681$
$1681 = 1681$

Success!