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Study Guide

Looking for patterns in sequences of numbers can be something you do because it's fun, or it can be something you do to get better at finding patterns in other contexts.

Being able to recognize and describe patterns is helpful in almost every area of thought imaginable. To support that statement, here's a selection of wildly different examples.

To decipher the encrypted message

WKH IRA OLNHV WR HDW WKH GXFNV

it's useful to recognize that the pattern WKH occurs twice. There aren't many three-letter words that would occur twice in that short of a message.

A thirteen year old kid made a breakthrough in solar power by recognizing an occurrence of the Fibonacci sequence in nature.

To determine whether a poem is a haiku, sonnet, limerick, or something else, you need to recognize what pattern it follows. If you write a good geeky haiku, you can win stuff.

### I Like Abstract Stuff; Why Should I Care?

It's possible to arrive at the same sequence of numbers from different directions by solving different problems.

### Sample Problem

How many ways are there to arrange

*n*pairs of parentheses in a way that makes grammatical sense? Something like)(

doesn't make grammatical sense.

Answer.

When

*n*= 1 there's only one way to arrange them:()

When

*n*= 2 there are two ways:()(), (())

When

*n*= 3 there are five ways:()()(), ((())), (())(), ()(()), (()())

Let

*C*= number of ways to grammatically arrange_{n}*n*pairs of parentheses.The numbers

*C*for_{n}*n*≥ 0 are called the*Catalan numbers*.### Sample Problem

How many ways are there to get from the point (0,0) to the point (

*n*,*n*) if every step is either up one or to the right one and we can't go over the line*y*=*x*?Answer. When

*n*= 1 there's only one way. We go right 1, then up 1:When

*n*= 2 there are 2 ways:When

*n*= 3 there are 5 ways:In general, there are

*C*ways. These are the same Catalan numbers_{n}*C*we saw with the parentheses example._{n}Catalan numbers count a lot of things, some of which don't seem to have anything to do with each other. It's tricky to come up with the formula from scratch, but the

*n*^{th}Catalan number is given byFor more information, you can look up the Catalan Numbers on the On-Line Encyclopedia of Integer Sequences here.

### How to Solve a Math Problem

There are three steps to solving a math problem.

- Figure out what the problem is asking.

- Solve the problem.

- Check the answer.

### Sample Problem

Some of the world's most expensive television sets cost $140,000 each.

If you deposit $400 in the bank and earn 10% interest every year, how many years will it be before you're able to buy this television set?

Answer.

1. Figure out what the problem is asking.

The problem is describing a geometric sequence. If you earn 10% interest per year, each year you'll have 1.1 times as much money as you did the previous year.

After one year you'll have

400(1.1),

after two years you'll have

400(1.1)(1.1),

and after

*n*years you'll have*a*= 400(1.1)_{n}.^{n}We want to know what value of

*n*makes*a*≥ 140,000._{n}2. Solve the problem.

Solve the inequality.

140,000 <

*a*_{n}140,000 < 400(1.1)

^{n}350 < (1.1)

^{n}log

_{1.1}350 <*n*61.46 <

*n*It would take 62 years before you had enough money to buy the television.

3. Check the answer.

We'll check the answer by finding

*a*_{61}and*a*_{62}.After 61 years, the amount of money you have is

*a*_{61}= 400(1.1)^{61}= 133,971.92.That's not quite enough. After 62 years you have

*a*_{62}= 400(1.1)^{62}= 147,369.11.That's plenty.

- Figure out what the problem is asking.

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