# Three numbers are in A.P. Their sum is 27 and the sum of their squares is 275. Find the numbers.

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

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