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

1 Answer
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:

(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!