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