Find the four numbers in A.P. whose sum is 20 and the sum of whose squares is 120.

Explanation

Let the required numbers be (a-3d), (a-d), (a+d) and (a+3d).
Then, `(a-3d) + (a-d) + (a+d) + (a+3d) = 20`
`rArr 4a = 20 rArr a = 5.`
`"And," (a-3d)^(2) + (a-d)^(2) + (a+d)^(2) + (a+3d)^(2) = 120`
`rArr 4(a^(2) +5d^(2)) = 120`
`rArr (a^(2) +5d^(2)) = 30 rArr 25 + 5d^(2) = 30 " " [because a = 5]`
`rArr 5d^(2) = 5 rArr d^(2) = 1 rArr d = +-1`
Thus, a = 5 and `d = +-1.`
Hence, the required numbers are (2, 4, 6, 8) or (8, 6, 4, 2).