Question

The sum of three terms which are in an AP is 33, if the product of first term and third term exceeds the second term by 29 find the AP.

In the solution of this question, Why do we take the three terms be a-d, a, a+d ?

Please Explain Me!

Answer 1

we take a-d a and a+d as they have a common difference which is d and it is more convenient to add as ' d ' gets cancelled out

a = 1 d = 2

a-d = -1
a = 1
a+d = 3

these three terms form an AP with 2 as the common difference

Answer 2

Note: We can take any value.

Let the three terms be a,a+d,a+2d.

Given that the sum of three terms is 33.

a + a + d + a + 2d = 33

3a + 3d = 33

a + d = 11 

a = 11 - d ----------- (1)


Given that product of 1st term and 3rd term exceeds the second term by 29.

a(a + 2d) = 29 + (a + d)

11 - d(11 - d + 2d) = 29 + (11 - d + d)

11 - d(11 + d)  = 29 + 11

121 - d^2 = 40

-d^2 + 121 - 40 = 0

81 - d^2 = 0

(9 + d)(9 - d) = 0

d = 9 (or) -9.


When d = 9:

a = 11 - 9

   = 2


a + d = 2 + 9

          = 11


a + 2d = 2 + 2(9)

            = 2 + 18

            = 20.



Therefore the AP is 2,11,20.


When d = -9:

a = 11 - (-9)

   = 20


a + d = 20 - 9

         = 11


a + 2d = 20 - 2(9)

            = 2



Therefore the AP is 2,11,20 (or) 20,11,2.



Hope this helps!