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The sum of the first three numbers in an Arithmetic Progression is 18. If the product of the first and the third term is 5 times the common difference, find the three numbers. - Mathematics

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Sum

The sum of the first three numbers in an Arithmetic Progression is 18. If the product of the first and the third term is 5 times the common difference, find the three numbers.

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Solution

Let the first three terms be a-d, a, a+d
We have been given that the sum of the first three terms of an A.P is 18
Equation becomes

a - d + a + a + d = 18

3a = 18

⇒ a = 6

Also, we have given the product of first and third term is 5 times the common difference

(a - d) (a + d) = 5d

a2 - d2 = 5d

⇒ a2 = 5d + d2 ................(∵ a = 6)

⇒ d2 + 5d = 36

⇒ d2 + 5d - 36 = 0

d2 + 9d - 4d - 36 = 0

⇒ d (d + 9) - 4 (d + 9) =0

⇒ (d - 4) (d + 9) = 0

⇒ d = 4, -9

When d = 4

First three numbers will be 6 -4, 6, 6+4
⇒ 2, 6, 10
When d= - 9
First three numbers will be 6 - (-9), 6, 6+ (-9)
⇒ 15, 6, -3

Concept: Sum of First n Terms of an AP
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