Final answer:
The arithmetic progression with the sum of the first n terms given by 4n - 5 is the correct one, with the first term of -1/2 and a common difference of 1/2. The 31st term of this AP is 15.
Step-by-step explanation:
The question involves finding the 31st term of an arithmetic progression (AP) where the sum of the first n terms is given by four different expressions. To determine which of the expressions belongs to an AP, we look for the sequence whose sum can be expressed in the form of n². From the provided information, we know that if we manipulate an AP's first and last terms by reciprocally adding and subtracting n - 1, n - 3, etc., we get a sequence where every term is equal to n, resulting in 2n² for the sum.
Matching the given options against this information, we identify that Option A, 2n - 3, does not resemble the form n², as such a sequence would not have the constant sum corresponding to an AP's sum formula. Option C, 4n - 5, correlates with the derived sum of 2n² when n is 3. Knowing this, we can deduce that the first term (a1) is ½(4 - 5) = -½, and the common difference (d) is also ½, as the difference between terms in an AP is consistent.
To find the 31st term, we use the formula an = a1 + (n - 1)d, resulting in a31 = -½ + (31 - 1)²/2 = 15. Therefore, the 31st term of the AP is 15.