Question

Find four numbers in Ap whose sum is 20 and the sum of whose square is 180

Answer 1

x - 3y, x-y, x+y, x+3y

x - 3y + x-y + x+y + x+3y = 20

4x = 20

x = 5

(x-3y)² + (x-y)² + (x+y)² + (x+3y)² = 180

x² - 6xy + 9y² + x² - 2xy + y² + x² + 2xy + y² + x² + 6xy + 9y² = 180

4x² + 20y² = 180

100 + 20y² = 180

y² = 80/20 = 4

y = ⁺/⁻2

so numbers

5 - 3(⁺/⁻2) = ⁻1 or 11

5 - (+/⁻2) = 3 or  7

5+(⁺/⁻2) =7  or   3

5+3(⁺/⁻2) = 11 or ⁻1