Sam decides to start a rumor. Sam tells the rumor to her three friends. Each of Sam's three friends then tells the rumor to three friends who have not heard the rumor. This continues for five total cycles. Sam telling her three friends was the first cycle. How many people, not including Sam, will have heard the rumor when the fifth cycle is complete?

Guest Aug 28, 2020

#1**-1 **

Why Sam decided to start a rumor? Nobody knows...

**Method 1: Construct a tree**

Not only are trees helpful to the environment, they are also great at solving math (though tedious for larger problems). Here is my "tree"

Sam

/ | \

/ | \ / | \ / | \

And so on.....

You'd eventually get **243** lines, but I wouldn't want to type that many lines....

**Method 2: Multiply**

Cycle 1: 3 = 3

Cycle 2: 3 * 3 = 9

Cycle 3: 3 * 3 * 3 = 27

Cycle 4: 3 * 3 * 3 * 3 = 81

Cycle 5: 3 * 3 * 3 * 3 * 3 = **243**

So yea, basically 3^5.

:)

ilorty Aug 28, 2020

#2**-1 **

Sam decides to start a rumor. Sam tells the rumor to her three friends. Each of Sam's three friends then tells the rumor to three friends who have not heard the rumor. This continues for five total cycles. Sam telling her three friends was the first cycle. How many people, not including Sam, will have heard the rumor when the fifth cycle is complete?

Cycle People Told Integer Format

1 3^{1} 3

2 3^{2} 9

3 3^{3} 27

4 3^{4} 81

5 3^{5} 243

Notice that the number of the cycle is the exponent of three that equals the number of people told.

_{.}

Guest Aug 28, 2020