To find the three numbers in arithmetic progression (AP) whose sum is 18, we start by denoting the three numbers as:
Let the three numbers be a – d, a, and a + d, where a is the middle term, and d is the common difference.
The sum of these three numbers can be expressed as follows:
(a – d) + a + (a + d) = 3a = 18
From this equation, we can solve for a:
3a = 18
a = 18 / 3
a = 6
Now, substituting a back into our expressions for the three numbers, we have:
- First number: a – d = 6 – d
- Second number: a = 6
- Third number: a + d = 6 + d
Next, we know that the product of the first and third numbers is given by:
(6 – d)(6 + d)
Using the difference of squares, this expands to:
36 – d2
Thus, if we need the product to be equal to some known value P, we can form the equation:
36 – d2 = P
To find the exact values of d and the three numbers, we would need a specific value for P. However, we’ve established that the center of our arithmetic progression is 6 and that the numbers are expressed in terms of d.
In conclusion, the three numbers in AP whose sum is 18 are:
First number: 6 – d, Second number: 6, Third number: 6 + d.
By choosing different values of d, you can generate countless sets of three numbers that satisfy the conditions laid out in this problem.