Question

The sum of the squares of five consecutive natural numbers is 1455. Find the numbers.

Answer 1

The five consecutive natural numbers are 15, 16, 17, 18, and 19.

Let the middle number be n. Since the numbers are consecutive natural numbers, the five numbers can be represented as:

(n – 2), (n – 1), n, (n + 1), and (n + 2)

(n – 2)² + (n + 1)² + n² + (n + 1)² + (n + 2)² = 1455

(n² – 4n + 4) + (n² – 2n + 1) + n² + (n² + 2n + 1) + (n² + 4n + 4) = 1455

5n² + 10 

= 1455

5n² = 1455 – 10

5n² = 1445

n² = `1445/5`

n² = 289

`n = sqrt289`

n = 17

Find the five numbers:

Substituting n = 17 back into our expressions

n – 2 = 17 – 2 = 15

n – 1 = 17 – 1 = 16

n = 17

n + 1 = 17 + 1 = 18

n + 2 = 17 + 2 = 19

Verification:

15² + 16² + 17² + 18² + 19²

= 225 + 256 + 289 + 324 + 361

= 1455