Wednesday, July 20, 2011

The Problem of the Week Day 3: Week of 7/17 - 7/23

Today, everybody get into sequence mode! We will have a lot of fun with these problems! Good luck.

Easy Problem: In order to find the next number in a simple sequence, look and see what you are adding/subtracting to get to the next number. If you find that they are the same, then you should be able to find the next number in the sequence. If they are not, see if you are multiplying or dividing by something to reach the next number. If so, you can also reach the next number in the sequence. That is called a geometrical sequence.

1) Plug z and b into this sequence:

z, b, 29, ___, 55, 68, 81, 94 ...

2) Look for a pattern in this sequence. It shouldn't be hard to find.

3) Determine what the number in the blank is by using this pattern. We will call that number n.

n = ___

Hard Problem: Here, we will solve our system and create an equation. Last month, we learned how to solve systems. If you remember that, you can eliminate the b's or c's by using "The Elimination Method."  Just to review, if you have two variables that are the same and have the same coefficient (number to the left of the variable that is multiplied by the variable), you can subtract both equations from each other to create a different equation with different variables. If you have an equation with three unknowns (a, b, and c), create two equations with this method, and then solve for that system. Then, plug those answers into an equation from the original system to get your third. Don't forget, a three-unknowns system requires three equations. If you only had two, create a third one.

1) Solve the system created from yesterday. If you have two variables, they should be m and b, and three variables is a, b, and c.

Linear Answer (if that was your system):
m = ___
b = ___

Quadratic Answer (if that was your system):
a = ___
b = ___
c = ___

2) Plug these answers into y = mx + b or y = ax^2 + bx + c to find your equation. This equation should determine any number in the system.

3) Just for fun, you should try figuring out the next number in the sequence, or maybe the spot for your favorite number. You can even find a fractional, negative, or imaginary value in the sequence. Or, you can see what spot is your favorite number by solving a two-step equation or using completing the square, factoring or the quadratic formula. However, this is optional.

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