Final answer:
To find three numbers in arithmetic progression whose sum is 24 and whose product is 312, we can use the formula for the sum of an arithmetic series and the equation for the product of the numbers.
Step-by-step explanation:
To find three numbers in arithmetic progression whose sum is 24 and whose product is 312, we can start by assigning variables to the numbers. Let the first number be 'a', and let the common difference between the numbers be 'd'.
Using the formula for the sum of an arithmetic series, we can express the three numbers:
a + (a + d) + (a + 2d) = 24
Expanding and simplifying the equation, we get:
3a + 3d = 24
Dividing both sides of the equation by 3, we have:
a + d = 8
Now, we can express the three numbers in terms of 'a' and 'd':
a, a + d, a + 2d
To find the product of these three numbers, we can substitute them into the equation:
a(a + d)(a + 2d) = 312
Expanding and simplifying the equation, we get:
a^3 + 3a^2d + 2ad^2 = 312
This equation can be solved by trial and error, or by using numerical methods such as factoring or graphing.