Question

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

Answer 1

Step-by-step explanation:

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

⇒x2 + x2 + 2x + 1 +x2 + 4x + 4 + x2 + 6x + 9 + x2 + 8x + 16 = 1455

⇒ 5x2 + 20x + 30 = 1455

⇒ 5(x2 + 4x + 6) = 1455

⇒ x2 + 4x + 6 = 291

⇒ x2 + 4x – 285 = 0

On splitting the middle term 4x as 19x – 15x, we get:

x2 + 19x – 15x – 285 = 0

⇒ x(x + 19) – 15(x + 19) = 0

⇒ (x + 19)(x– 15) = 0

⇒x + 19 = 0 or x – 15 = 0

⇒ 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

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