Final answer:
To find the total number of cartons needed for the complete display, we need to find the sum of the cartons in each row of the triangle. Using the formula for the sum of an arithmetic series, the total number of cartons needed is 120.
Step-by-step explanation:
In this question, the grocery clerk is setting up a display of 12-pack cartons of cola in the shape of a triangle. The base of the triangle has 15 cartons and there is one carton at the top. To find the total number of cartons needed for the complete display, we need to find the sum of the cartons in each row of the triangle. Starting with the base, there are 15 cartons. Moving up to the next row, there are 14 cartons. Continuing this pattern, we find that there are 15 + 14 + 13 + 12 + ... + 3 + 2 + 1 cartons. This is an example of an arithmetic series, which can be calculated using the formula Sn = (n/2)(a1 + an), where Sn is the sum of the series, n is the number of terms, a1 is the first term, and an is the last term. In this case, n is 15, a1 is 15, and an is 1. Plugging these values into the formula, we get Sn = (15/2)(15 + 1) = (15/2)(16) = 120 cartons. Therefore, the correct answer is D. 120 cartons.