Question

Three numbers in A.p have sum 18 and their sum of their squares is 180.find the numbers

Answer 1

I hope my answer ia correct

Answer 2

Let three numbers of AP be a+d, a ,and a-d

now according to question
sum of these terms is 18 so,
= (a+d)+(a)+(a-d)= 18

= a+d+a+a-d= 18
= 3a= 18
a= 18/3
= a= 6........(1)

second situation given in question
sum their squares is 180
(a+d)²+(a)²+(a-d)²= 180
= (a²+d²+2ad)+(a²)+(a²+d²-2ad)= 180
= a²+d²+2ad+a²+a²+d²-2ad=180
=3a²+2d²=180
= 3(6)²+2d²=180.......(putting a=6 from (1))
= 108+2d²= 180
=2d²= 180-108
2d²= 72
= d²= 36
d= +6 or -6

So there are 2 sets of three terms
first where a= 6 and d= -6
putting values in a+d, a, and a-d
three terms= 0,6,12

second when a= 6 and d=+6
putting values in a+d, a, and a-d
three terms= 12, 6, 0

hope this helps you