Sequences and Series Word Problem

I caught a cold five days ago. Each day after I caught it, I spread it to three new people, as did anyone who caught the cold (beginning the day after they catch it, they spread it to three new people each day). Assuming I was the first person to catch the cold, no one has gotten over the cold, and that no person was infected from two different sources, how man people will have the cold by the end of today.

Re: Sequences and Series Word Problem

Hey Espionage.

Can you show us what you have tried? What mathematical/statistic models do you think best represent your problem?

Re: Sequences and Series Word Problem

I don't have any mathematical model but I do have a number tree or a geometric representation. I really don't understand the problem very clearly but I tried:

1 + 4 + 12 + 36 + 108. A geometric progression with common ratio 3 and a(1)=1 to model only one person getting the flu. But it really doesn't look right.

Re: Sequences and Series Word Problem

I just have to ask: can the people with a cold only spread it to three other people once or can they keep spreading it to three other people every single day?

Re: Sequences and Series Word Problem

Quote:

Originally Posted by

**Espionage** I don't have any mathematical model but I do have a number tree or a geometric representation. I really don't understand the problem very clearly but I tried:

1 + 4 + 12 + 36 + 108. A geometric progression with common ratio 3 and a(1)=1 to model only one person getting the flu. But it really doesn't look right.

Your calculations are wrong. Only you had the cold on the first day- 1. The second day you gave it to 3 other people and still had it yourself so 4 people had the cold. On the third day each of you four gave it to three new people, a total of 4(3)= 12 new people and still had it yourselves: 4+ 12= 16 people had the cold on the third day. On the fourth day, all 16 gave it to 3 new people so 16(3)= 48 and still had it your selves: 48+ 16= 64. On the fifth day, all 64 gave to 3 new people so 64(3)= 192 and still had it them selves: 192+ 64= 256. (Since we are adding the people who already had the cold into the value each day, this is a **sequence**, not a series. There should be no "+" between the numbers.)

It should be easy to see a pattern (for one thing 1, 4, 16, 64, and 256 are all powers of 2) and then use, say, induction to prove that pattern is correct.

Perhaps you were trying to calculate the number of new people who caught the cold each day and add them. But, if so you did that wrong as well. The first day there is 1, the second day 3(1)= 3 new, the third day 3(1+3)= 3(4)= 12, but the fourth day 3(12+3+ 1)= 4(17)= 68, not 64.

Re: Sequences and Series Word Problem

Hello, Espionage!

I got the same answer as HallsofIvy by brute-force listing.

Quote:

I caught a cold five days ago.

Each day after I caught it, I spread it to three new people, as did anyone who caught the cold

(beginning the day after they catch it, they spread it to three new people each day).

Assuming I was the first person to catch the cold, no one has gotten over the cold,

and that no person was infected from two different sources,

how many people will have the cold by the end of today?

$\displaystyle \begin{array}{ccccccc}\text{Day 1:} & \text{Only you had a cold.} && &&1 \\ \text{Day 2:} & \text{You gave the cold to 3 others.} && 1+3 &=& 4 \\ \text{Day 3:} & \text{The 4 of you gave the cold to }4(3) = 12\text{ people.} && 4+12 &=& 16 \\ \text{Day 4:} & \text{The 16 of you gave the cold to }16(3) = 48\text{ people.} && 16+48 &=& 64 \\ \text{Day 5:} & \text{The 64 of you gave the cold to }64(3) = 192\text{ people.} && 64 + 192 &=& 256 \\ \text{Day 6:} & \text{The 256 of you gave the cold to }256(3) = 768\text{ people.} && 256 + 768 &=& \boxed{1024} \end{array}$

$\displaystyle \text{On the }n^{th}\text{ day, the number of infected people is: }\:4^{n-1}$