Question

Three numbers are in A.P. If the sum of these numbers is 27 and the product 648, find the numbers.

Answer 1

In the given problem, the sum of three terms of an A.P is 27 and the product of the three terms is 648. We need to find the three terms.

Here,

Let the three terms be (a - d), a, (a + d) where a is the first term and d is the common difference of the A.P

So,

(a - d) + a(a + d) = 27

3a = 27

a = 9 ......(1)

Also

(a - d)a(a + d) = a + 6

`a(a^2 - d^2) = 648`      [Using `a^2 - b^2 = (a +              b)(a - b)`]

`9(9^2 - d^2) = 648`

`81 - d^2  = 72`

Further solving for d

`81 - d^2 =72`

`81 - 72 = d62`

`81 - d^2 = 72`

Further solving for d

`81 - d^2 = 72`

`81 - 72 = d^2`

`d = sqrt9`

d = 3....(2)

Now, substituting (1) and (2) in three terms

First term = a - d

So, a - d = 9 - 3

= 6

Also

Second term  = a

So,

a= 9

Also

Third term = a + d

So

a + d = 9 + 3

= 12

Therefore the three term are 6, 9 and 12