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The sum of the squares of two consecutive natural numbers is 113. Find the numbers.
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