The sum of three numbers in A.P. is 27 and the sum of their squares is 293. Find the numbers.

Explanation

Let the three numbers in A.P. be a - d, a,a+ d.
Given , ` a - d + a+ a + d = 27 `
`rArr 3a = 27`
`rArr a = 9 `
Also , `(a - d)^(2) + a^(2) + (a + d)^(2) = 293`
`rArr a^(2) + d^(2) - 2ad + .a^(2) + a^(2) + d^(2) + 2ad = 293 `
`rArr 3a^(2) + 2d^(2) = 293`
`rArr 3(9)^(2) + 2d^(2) = 293`
`rArr 2d^(2) = 293 - 243`
`rArr 2d^(2) = (50)/(2) = 25 `
`rArr d = pm5 `
If d = - 5, the three numbers are 14, 9, 4
If d = 5 , the three nubers are 4, 9 , 14 .