Question

Find 4 numbers in A.P. whose sum is 4 and sum of whose squares is 84.

Answer 1

a − 3d, a − d, a + d, a + 3d

a − 3d + a − d + a + d + a + 3d = 4

4a = 4

a = `4/4`

a = 1

(a − 3d)² + (a − d)² + (a + d)² + (a + 3d)² = 84

(1 − 3d)² + (1 − d)² + (1 + d)² + (1 + 3d)² = 84

1 + 9d² − 6d + 1 + d² − 2d + 1 + d² + 2d + 1 + 9d² + 6d = 84

20d² + 4 = 84

20d² = 84 − 4

20d² = 80

d² = `80/20`

d² = 4

d = +2, −2

a = 1 and  d = 2 Or a = 1 and  d = −2

If d = 2 and a = 1:

a − 3d, a − d, a + d, a + 3d

⇒ a − 3d

=1 − 3(2)

= 1 − 6

= −5

⇒ a − d

= 1 − 2

= −1

⇒ a + d

= 1 + 2

= 3

⇒ a + 3d

= 1 + 3(2)

= 1 + 6

= 7

The four numbers are −5, −1, 3, and 7.

If d = −2 and a = 1:

a − 3d, a − d, a + d, a + 3d

⇒ a − 3d

=1 − 3(−2)

= 1 + 6

= 7

⇒ a − d

= 1 − (−2)

= 3

⇒ a + d

= 1 + (−2)

= −1

⇒ a + 3d

= 1 + 3(−2)

= 1 − 6

= −5

The four numbers are 7, 3, −1, and −5.