The sum of the squares of three consecutive natural numbers is 149. Find the numbers.

Explanation

Let three consecutive natural numbers be `x,(x+1)and(x+2)` respectively.
According to given condition
`x^(2)+(x+1)^(2)+(x+2)^(2)=149`
`impliesx^(2)+(x^(2)+1+2x)+(x^(2)+4+4x)=149`
`implies3x^(2)+6x+5=149`
`implies3x^(2)+6x-144=0`
`impliesx^(2)+2x-48=0`
`impliesx^(2)+8x-6x-48=0`
`impliesx(x+8)-6(x+8)=0`
`implies(x+8)(x-6)=0`
`impliesx=-8andx=6`
`impliesx=6" "(x=-8"is not a natural number")`
Hence, the required natural number are `6,(6+1),(6+2)=6,7,8` respectively.