# Thread: Exponential Function For Situations

1. ## Exponential Function For Situations

What is the equation for this word problem? How can you find it?

Professor Livingstone loves to perform acts of kindness. He decides to conduct an experiment in which he performs an act of kindness for only two students. He then requests that these students perform an act of kindness for three new students every day. Each new student who receives an act of kindness is given instructions to pay it forward to three additional students every day. Assuming that each student is a recipient only once and everyone who has received an act of kindness continues to fulfill three acts of kindness each day, how long will it take for the entire student population of 6100 students to receive an act of kindness?

2. ## Re: Exponential Function For Situations

I would start by calculating specific values for a few days writing the number of "acts of kindness" on day t as N(t). The first day Professor Livingston performs 3 acts of kindness. N(0)= 3. The next day he and those 3 others each perform 3 acts of kindness: N(1)= 4(3)= 12. The next day he and the first three as well as the new 12, so 15 altogether, each perform three acts of kindness: N(2)= 16(3)= 48. The next he, the first three, the second 12, and the new 48, so a total of 1+ 3+ 12+ 48= 64, each perform 3 acts of kindness so N(3)= 3(64)= 192.

Each of those is obviously divisible by 3:N(0)= 3= 3(1), N(1)= 12= 3(4), N(3)= 48= 3(16), and N(3)= 192= 3(64). I notice that each of 1, 4, 16, and 64 are powers of 4! I conjecture that $\displaystyle N(m)= 3(4^m)$. Can you prove (perhaps by induction) that this formula is correct?

Given that, the total number of acts of kindness in m days is the sum: $\displaystyle 3+ 12+ 48+ \cdot\cdot\cdot+ 3(4^m)= 3(1+ 4+ 16+ 64+ \cdot\cdot\cdot+ 4^m)$. Tjat last is geometric series. Do you know the formula for the sum of a geometric series?