Question

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

Answer 1

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Answer 2

Answer: The required A.P. is,

2, 6, 10  or  15, 6, -3

Step-by-step explanation:

Let d be the common difference of the A.P.

Since, The sum of the first three numbers in an arithmetic progression is 18.

⇒ The middle term = Average of the numbers = 18/3 = 6

Hence, the first term of the A.P = 6 - d

And, the last term of the A.P. = 6 + d

According to the question,

The product of the first and the third term is 5 times the common difference

⇒ (6-d) (6+d) = 5d

⇒ 36 - d² = 5d

⇒ d² + 5d - 36 = 0

⇒ d² + 9d - 4d - 36 = 0

⇒ d(d+9) - 4(d+9)=0

⇒ (d-4)(d+9) = 0

⇒ d = 4 or -9

When d = 4,

Then the A.P. is,

6 - 4, 6, 6 + 4

⇒ 2, 6, 10.

When d = -9,

Then the A.P. is,

6 -(-9) , 6, 6 + (-9)

⇒ 15, 6, -3