Question
A grocery clerk sets up a display of 12 - pack cartons of soda. There are 15 cartons at the base of the triangle and one at the top. How many cartons of cola are needed for the complete display?
Answers
GPT-4o mini
The display described forms a triangular arrangement of cartons. The number of cartons in such a triangular formation can be determined using the formula for the \(n\)-th triangular number, which is given by:
\[
T_n = \frac{n(n + 1)}{2}
\]
where \(n\) is the number of cartons at the base of the triangle.
In this case, the base consists of 15 cartons. Thus, \(n = 15\). We need to calculate \(T_{15}\):
\[
T_{15} = \frac{15(15 + 1)}{2} = \frac{15 \times 16}{2} = \frac{240}{2} = 120
\]
Therefore, the total number of cartons needed for the complete display is:
\[
\boxed{120}
\]
\[
T_n = \frac{n(n + 1)}{2}
\]
where \(n\) is the number of cartons at the base of the triangle.
In this case, the base consists of 15 cartons. Thus, \(n = 15\). We need to calculate \(T_{15}\):
\[
T_{15} = \frac{15(15 + 1)}{2} = \frac{15 \times 16}{2} = \frac{240}{2} = 120
\]
Therefore, the total number of cartons needed for the complete display is:
\[
\boxed{120}
\]