Let the three numbers are in AP = a, a + d, a + 2d
According to the question,
The sum of three terms = 27
⇒ a + (a + d) + (a + 2d) = 27
⇒ 3a + 3d = 27
⇒ a + d = 9
⇒ a = 9 – d …(i)
and the sum of their squares = 275
⇒ a2 + (a + d)2 + (a + 2d)2 = 275
⇒ (9 – d)2 + (9)2 + ( 9 – d + 2d)2 = 275 [from(i)]
⇒ 81 + d2 – 18d + 81 + 81 + d2 + 18d = 275
⇒ 243 + 2d2 = 275
⇒ 2d2 = 275 – 243
⇒ 2d2 = 32
⇒ d2 = 16
⇒ d = √16
⇒ d = ±4
Now, if d = 4, then a = 9 – 4 = 5
and if d = – 4, then a = 9 – ( – 4) = 9 + 4 = 13
So, the numbers are →
if a = 5 and d = 4
5, 9, 13
and if a = 13 and d = – 4
13, 9, 5