To find the product of the first and third numbers in an arithmetic progression (AP) where the sum of three numbers is 18, we start by letting the three numbers be:
- First number = a – d
- Second number = a
- Third number = a + d
In an AP, the middle number is the average of the first and third, so:
Sum = (a – d) + a + (a + d) = 3a = 18
From this equation, we can solve for ‘a’:
3a = 18
a = 18 / 3 = 6
So, the three numbers can be expressed as:
- First number = 6 – d
- Second number = 6
- Third number = 6 + d
Now, to find the product of the first and third numbers:
Product = (6 – d)(6 + d)
This can be simplified using the difference of squares:
Product = 62 – d2
Calculating that becomes:
36 – d2
Since we don’t have the value of ‘d’, we can summarize that the product of the first and third numbers will be expressed as:
Product = 36 – d2
Thus, the product depends on the specific value of ‘d’, but in any case, the product of the first and third numbers is maximized when d=0, yielding a product of 36.