Question

Find your numbers in AP whose sum is 28 and the sum of whose squares is 216.

Answer 1

Let the required numbers be (a - 3d)(a - d), (a + d) and (a + 3d).
Then (a - 3d) + (a - d)​ + (a + d) + (a + 3d) = 28
⇒ 4a = 28
⇒ a = 7​

Also, (a - 3d)² + (a - d)² + (a + d)²​ + (a + 3d)² = 216
⇒ 4 (​a² + 5d²)​= 216
⇒ 4⨯​( 49​ + 5d²)​ = 216 ​
⇒ 5d² = 54 - 49 = 5
⇒ d² = 1​
⇒ d = $\pm$1
Thus, a = 7 and d = $\pm$1
Hence, the required numbers are (4, 6, 8, 10) or (10, 8, 6, 4).