SUM OF THE SQUARES OF TWO CONSECUTIVE NUMBERS - MATHS - 2025

To find out what is the sum of the squares of two consecutive numbers, a formula can be found, with which it is enough to substitute the numbers involved to obtain the result.

This formula can be found in a general way, that is, it can be used for any pair of consecutive numbers.

By saying "consecutive numbers," you are implicitly saying that both numbers are whole numbers. And by "the squares" he is referring to squaring each number.

For example, if the numbers 1 and 2 are considered, their squares are 1² = 1 and 2² = 4, therefore, the sum of the squares is 1 + 4 = 5.

On the other hand, if the numbers 5 and 6 are taken, their squares are 5² = 25 and 6² = 36, with which the sum of the squares is 25 + 36 = 61.

What is the sum of the squares of two consecutive numbers?

The goal now is to generalize what was done in the previous examples. To do this, it is necessary to find a general way to write an integer and its consecutive integer.

If you look at two consecutive integers, for example 1 and 2, you can see that 2 can be written as 1 + 1. Also, if the numbers 23 and 24 are observed, it is concluded that 24 can be written as 23 + 1.

For negative integers this behavior can also be verified. Indeed, if -35 and -36 are considered, it can be seen that -35 = -36 + 1.

Therefore, if any integer "n" is chosen, then the integer consecutive to "n" is "n + 1". Thus, a relationship between two consecutive integers has already been established.

What is the sum of the squares?

Given two consecutive integers "n" and "n + 1", then their squares are "n²" and "(n + 1) ²". Using the properties of notable products, this last term can be written as follows:

(n + 1) ² = n² + 2 * n * 1 + 1² = n² + 2n + 1.

Finally, the sum of the squares of the two consecutive numbers is given by the expression:

n² + n² + 2n + 1 = 2n² + 2n +1 = 2n (n + 1) +1.

If the previous formula is detailed, it can be seen that it is only enough to know the smallest integer "n" to know what the sum of the squares is, that is, it is only enough to use the smallest of the two integers.

Another perspective of the formula obtained is: the chosen numbers are multiplied, then the result obtained is multiplied by 2 and finally 1 is added.

On the other hand, the first addend on the right is an even number, and adding 1 will result in odd. This says that the result of adding the squares of two consecutive numbers will always be an odd number.

It can also be noted that since two numbers squared are being added, then this result will always be positive.

Examples

1.- Consider the integers 1 and 2. The smallest integer is 1. Using the previous formula, it is concluded that the sum of the squares is: 2 * (1) * (1 + 1) +1 = 2 * 2 + 1 = 4 + 1 = 5. Which agrees with the counts made at the beginning.

2.- If the integers 5 and 6 are taken, then the sum of the squares will be 2 * 5 * 6 + 1 = 60 + 1 = 61, which also coincides with the result obtained at the beginning.

3.- If the integers -10 and -9 are chosen, then the sum of their squares is: 2 * (- 10) * (- 9) + 1 = 180 + 1 = 181.

4.- Let the integers in this opportunity be -1 and 0, then the sum of their squares is given by 2 * (- 1) * (0) + 1 = 0 +1 = 1.

References

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4. Guevara, MH (nd). The Set of Whole Numbers. EUNED.
5. Oteyza, E. d. (2003). Albegra. Pearson Education.
6. Smith, SA (2000). Algebra. Pearson Education.
7. Thomson. (2006). Passing the GED: Mathematics. InterLingua Publishing.