Q9 of 85 Page 110

Three numbers in A.P. have the sum 18 and the sum of their squares is 180. Find the numbers in the increasing order.

We know that the sum of 3 numbers in the AP is 18.


Let’s say that those 3 numbers are: a – d, a, a + d


So we can say that, a – d + a + a + d = 18


⇒ 3a = 18


⇒ a= 6


Now, we have that the sum of squares of these 3 numbers is 180.


So we can say that, (a – d) 2 + a2 + (a + d) 2 = 180


⇒ a2 – 2ad + d2 + a2 + a2 + 2ad + d2 = 180


⇒ 3a2 + 2d2 = 180


We know that a = 6,


So, 3(6)2 + 2d2 = 180


⇒ 3(36) + 2d2 = 180


⇒ 108 + 2d2 = 180


⇒ 2d2 = 180 – 108


⇒ 2d2 = 72


⇒ d2 = 36


⇒ d = 6


Now, our 3 numbers a – d, a, a + d are 6 – 6, 6, 6 + 6 = 0, 6, 12 respectively.


∴ The 3 numbers of the A.P in the increasing order are : 0, 6, 12.


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