A rumor about the school dance is being spread throughout a school. It began at 8 a.m. this morning, and each hour the fraction of the students that know the rumor triples. The principal does not know what fraction of students started the rumor.

1. What is the multiplier for the geometric sequence in this situation?

   Hint (a):

   How do you triple something?

2. By 3 p.m. this afternoon, every student has heard the rumor. What fraction of students had heard the rumor at 2 p.m.?

   Hint (b):

   Between 2 p.m. and 3 p.m., the number of people who had heard the rumor tripled so that the whole student body knows.
   If the whole student body is $1$, what fraction times $3$ equals $1$? That is, $? · 3 = 1$?

3. Let $a$ represent the unknown fraction of the student population that started the rumor. Write an equation for a geometric sequence that represents the fraction of students that have heard the rumor after $n$ hours.

   Answer (c):

   $a · 3^{n} = y$

4. Use the fact that the every student has heard the rumor by 3 p.m. and your equation to determine the fraction of the student population that started the rumor.

   Step 1(d):

   Use the equation from part (c) where n is the number of hours that have passed between 8 a.m. and 3 p.m. and y is 1.

   Step 2(d):

   Solve the equation for $a: a · 3^{7} = 1$

5. If a group of $3$ students started the rumor initially, how many students are in the school?

   Hint (e):

   Let s = the total number of students in school.

   $\text{Since }a = \frac{1}{2187}, \frac{3}{s}\text{ is an equivalent fraction. That is, }\frac{1}{2187}=\frac{3}{s}.$