Question

Four different integers form an increasing A.P. . One of these numbers is sum of square of other three numbers. Find the numbers

Answer 1

four different integers form an increasing A.P

we can assume four different integers are ; a - 3d , a -d, a + d , a + 3d , these are increasing in order.

so, a + 3d is biggest number.

now a/c to question,

one of these number is sum of square of other three numbers.

so, (a + 3d) = (a - 3d)² + (a - d)² + (a + d)²

= a² + 9d² - 6ad + a² + d² - 2ad + a² + d² + 2ad

= 3a² + 11d² - 6ad ......(1)

or, 3a² + 11d² - 6ad - a - 3d = 0

or, 3a² - a(6d + 1) + 11d² - 3d = 0

for a is real so, D = (6d + 1)² - 12(11d² - 3d) ≥ 0

or, 36d² + 12d + 1 - 132d² + 36d ≥ 0

or, -96d² + 48d + 1 ≥ 0

or, 96d² - 48d - 1 ≤ 0

we can choose d from hit and trial method.

if we choose d = 1/2 , 96 × 1/4 - 48 × 1/2 - 1 ≤ 0 satisfied

so, d = 1/2 is correct.

putting it equation (1), we get a = 1/2

so, a - 3d = 1/2 - 3/2 = -1

a - d = 1/2 - 1/2 = 0

a + d = 1/2 + 1/2 = 1

a + 3d = 1/2 + 3/2 = 2

so, numbers are -1, 0, 1 , 2