Question

Prove that in the sequence 1, 2, 3, ….. of natural numbers, 1 added to the product of any two alternatives numbers perfect square.

Answer 1

Let us assume a number a in the sequence of natural numbers 1, 2, 3,…

Alternate number after a = a + 2

Product of the two alternate numbers = (a + 2) × a

= a² + 2a

If one is added to this expression, it will become a² + 2a + 1.

Now, a² + 2a + 1 = a² + 2 × a × 1 + 1²

= (a + 1)² {(x + y)² = x² + y² + 2xy}

Thus, when 1 is added to the product of any two alternate natural numbers, the result obtained is a perfect square.