The sum of the squares of two consecutive natural numbers is 1513. What is the sum of these numbers?

Let us denote by a that of the two given numbers, which is the smallest.

Then that of the two given numbers, which is the largest, must be a + 1.

In the initial data for this task, it is reported that if you add the squares of these two numbers, the result will be 1513, therefore, we can make the following equation:

a² + (a + 1) ² = 1513,

solving which, we get:

a² + a² + 2a + 1 = 1513;

2а² + 2а + 1 – 1513 = 0;

a² + a – 756 = 0;

a = (-1 ± √ (1 + 3024)) / 2 = (-1 ± √3025) / 2 = (-1 ± 55) / 2;

a = (-1 + 55) / 2 = 54/2 = 27.

Let’s find the required amount:

a + a + 1 = 2a + 1 = 2 * 27 + 1 = 55.

Answer: the correct answer is   55.