1. The sum of the squares of two
consecutive natural numbers is 113. Find the numbers.
Sol. Let the two consecutive natural
numbers be x and x + 1
As per the given condition,
x2 + (x + 1)2
= 113
∴ x2
+ x2 + 2x + 1 = 113
∴ 2x2 + 2x + 1 – 113 = 0
∴ 2x2 + 2x – 112 = 0
Dividing by 2 we get,
x2 + x – 56 = 0
∴ x2 + 8x – 7x – 56 = 0
∴ x (x + 8) – 7 (x + 8) = 0
∴ (x + 8) (x – 7) = 0
∴ x + 8 = 0 or x – 7 = 0
∴ x = – 8 or x = 7
∵ x
is a natural number
∴ x cannot be negative
∴ x
= 7
And x + 1 = 7 + 1 = 8
∴ The two consecutive natural numbers are 7
and 8
respectively