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The sum of three numbers in AP is 18. If the product of first and third number is five times the common difference, then what are the first 3 numbers?

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3, 7, 11 or 16, − 9, − 3

4, 8, 12 or 17, − 9, − 2

2, 6, 10 or 15, 6, - 3

5, 3, 9 or 11, − 8, − 3

answer is C.

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Detailed Solution

Given that the sum of three numbers in AP is 18.
Let the numbers are

a − d, a, a + d

.
So, according to given condition,

a − d + a + a + d = 18 ⇒ 3 a = 18 ⇒ a = 6

Also, given that the product of the first and third numbers is five times the common difference.
So,

a − d × a + d = 5 d ⇒ 6 − d × 6 + d = 5 d ⇒ 62 − d 2 = 5 d [(a − b) (a + b) = (a 2 − b 2)] ⇒ d 2 + 5 d − 36 = 0

Solving the quadratic equation,

d 2 + 5 d − 36 = 0

⇒ d 2 + 9 d − 4 d − 36 = 0 ⇒ d d + 9 − 4 d + 9 = 0 ⇒ d + 9 d − 4 = 0 ⇒ d = 4, − 9

Therefore, the common difference is either

4 or -9

.
Thus, the numbers when the common difference is 4,

= a − d, a, a + d = 6 − 4, 6 6 + 4 = 2, 6, 10

Also, the numbers when the common difference is

− 9

= (a - d), a, (a + d)

= (6 - (- 9)), 6, (6 - 9)

= 15, 6, - 3

Therefore, the first three number will be either

2, 6, 10 or 15, 6, - 3

.
Hence, option (3) is correct.

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