let the four parts be
a, a+d, a+2d, and a+3d
a + a+d + a+2d + a+3d= 24
4a + 6d = 24
2a + 3d = 12 ----> d = (12-2a)/3
a^2 + (a+d)^2 + (a+2d)^2 + (a+3d)^2 = 164
4a^2 + 12ad + 14d^2 = 164
2a^2 + 6ad + 7d^2 = 82
2a^2 + 6a(12-2a)/3 + 7(12-2a)^2/9 = 82
times 9
18a^2 + 216a - 36a^2 + 1008 - 336a + 28a^2 = 738
10a^2 -120a + 270 = 0
a^2 - 12a + 27 = 0
(a-9)(a-3) = 0
a = 9 or a = 3
if a=9, d = -2, AP is 9, 7, 5, and 3
if a=3, d = 2 , the AP is 3,5,7, and 9
(notice the numbers are just reversed)
Divide 24 into four parts which are in A.P. such that the sum of their squares is 164.
1 answer
1 answer