Wednesday, August 17, 2011

The Problem of the Week Day 3: Week of 8/14 - 8/20

Today, we will finish up on the chess tournament problem. You should have already noticed a pattern.

Easy Problem: There are two types of formulas you can have in a sequence. One of which is the explicit formula, which is a formula based on the nth term. For instance, the explicit formula for the sequence 2, 4, 6, 8, 10, ... would be An = 2n. If you wanted the 6th tern, you would plug 6 in for n to get 2(6) = 12.

The other type of formula is the recursive formula, which is based on the previous term. In the even number sequence, the recursive formula would be An = An-1 + 2 because it is the previous term plus two.

1) What is the explicit formula for the chess tournament problem?

2) What is the recursive formula for the chess tournament problem?

3) If there were b players in the tournament, how many games would it take to find a winner fairly?

Hint: Use the explicit formula for number three!

Hard Problem: At CTY, we used various strategies to determine explicit formulas. However, I tend to lean towards the method I taught last month, with the systems. Just to remind you how to find the system, you must first find your common differences. For the sequence 2, 4, 6, 8, 10, ..., the differences are in the first row. Therefore, you are dealing with a first-degree, or linear, equation. So, we take our base equation for this: An = mn + b. Then, plug in values for n and An to create the system. For instance, you would first plug 1 in for n and 2 in for An, and have the first equation, m + b = 2. Then, you would create a second one, 2m + b = 4, to get our constants, m = 2 and b = 0. This makes the explicit formula An = 2n + 0 which becomes An = 2n.

1) Find common differences in the Pizza Problem.

2) Create a system of equations.

If you want to get ahead, try solving the system, or even try finding a recursive formula.

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