Not counting the fact that students need to take food from the shelves: Assume that I donate $1 $ pound of food and I get $3 $ of my students to donate, but they only donate $35\% $ of what I gave. However, each of those students tell $3 $ of their friends to donate, and they are each able to give $35\% $ of what that student gave. If this pattern continues, represent it as a series. Does the series converge or diverge? What does this mean for the food shelf?
Please help me with question. It is quite confusing. Question says "I get $3 $ of my students to donate, but they only donate $35\% $ of what I gave." So did each student donated $0.35$ pound or $3$ of them combined donated $0.35$?
Assuming that the pattern is shown by "However, each of those students tell $3$ of their friends to donate, and they (the three friends) are each able to give $35\%$ of what that student gave", the number of people donating would be $3^n$ where $n$ is the number of steps, and the amount donated by each person in step $n$ would be $0.35^n$.
Hence the series (stated by @Ananya in the comments) would be: $$1+3^10.35^1+3^20.35^2+3^30.35^3+\dots=\sum_{i=0}^n\left(3\times0.35\right)^n=\sum_{i=0}^n1.05^n$$
Assuming that this pattern actually happens, the most realistic thing would be that the food shelf would run out of space, as the series is divergent. What really happens to the food shelf may be quite open-ended depending on the nature of the shelf.