Question

the sum of the first three no. in an A.P is 18. if the product of the first and third term is 5 times the common difference , find the three no.

Answer 1

Answer:

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Answer 2

Answer:

3 terms are 15 , 6 , -3 or 2 , 6 , 10.

Step-by-step explanation:

Given: Sum of first three terms of an AP = 18

           Product of 1st and 3rd term = 5 times common difference

To find: three nos.

Let say the first three terms of an AP are a - d , a , a + d

The first term of AP = a - d

And common difference = 2nd term - 1st term = a - ( a - d )

                                         = a - a + d = d

Sum of first three term of AP = 18

⇒ ( a - d ) + ( a ) + ( a + d ) = 18

    a - d + a + a + d = 18

    3a = 18

    a = 6

now, Product of 1st and 3rd term = 5  times common difference

⇒ $a_1\,\times\,a_3\:=\:5\times d$

   ( a - d ) × ( a + d ) = 5 × d

   a² - d² = 5d     ( using identity, ( x - y )( x + y ) = x² - y² )

   6² - d² = 5d

   d² + 5d - 36 = 0

   d² + 9d -4d - 36 = 0

   d ( d + 9) - 4 ( d + 9 ) = 0

   ( d + 9 ) ( d - 4 ) = 0

⇒ d = - 9 , 4

when d = -9

⇒ a - d = 6 - (-9) = 15 , a = 6 , a + d = 6 + (-9) = -3

when d = 4

⇒ a - d = 6 - (4) = 2 , a = 6 , a + d = 6 + (4) = 10

 Therefore, 3 terms are 15 , 6 , -3 or 2 , 6 , 10.