Final answer:
The number of terms in the arithmetic progression that sums to 252, with a first term of -16 and a last term of 72, is 9.
Step-by-step explanation:
The question asks to find the number of terms (n) in an arithmetic progression with a given sum, first term, and last term. The relevant information needed to solve this problem involves understanding the sum of an arithmetic sequence and how it is related to the number of terms, the first term, and the last term.
The sum (S) of an arithmetic progression with n terms, first term a, and common difference d can be given by S = n/2 * (2a + (n-1)d). In this case, the sum is 252, the first term (a) is -16, and the last term (l) is 72. We can also use the formula S = n/2 * (a + l), where l is the last term, since in an arithmetic progression the average of the first and last term gives the mean of the series. Using the second formula we can set up the equation 252 = n/2(-16 + 72).
Solve this equation to find n.
252 = n/2 * (56),
252 = 28n,
n = 252/28,
n = 9.
The arithmetic progression has 9 terms.