Question

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

Answer 1

Let the five consecutive natural numbers be x, (x + 1), (x + 2), (x + 3) and (x + 4).
By the given condition, we get:
x² + (x + 1)² + (x + 2)² + (x + 3)² + (x + 4)² = 1455
$\Rightarrow$x² + x² + 2x + 1 +x² + 4x + 4 + x² + 6x + 9 + x² + 8x + 16 = 1455
$\Rightarrow$ 5x² + 20x + 30 = 1455
$\Rightarrow$ 5(x² + 4x + 6) = 1455
$\Rightarrow$ x² + 4x + 6 = 291
$\Rightarrow$ x² + 4x – 285 = 0
On splitting the middle term 4x as 19x – 15x, we get:
x² + 19x – 15x – 285 = 0
$\Rightarrow$ x(x + 19) – 15(x + 19) = 0
$\Rightarrow$ (x + 19)(x– 15) = 0
$\Rightarrow$x + 19 = 0 or x – 15 = 0
$\Rightarrow$ x = –19 or x = 15
Since x is a natural number, which cannot be negative, x = 15
Thus, the five consecutive numbers are 15, 16, 17, 18 and 19