In the next few days, we will be looking at a topic called "Sequences and Pattern Recognition," which is basically finding the next number in a pattern. In first grade, you make the little bumps with a plus two on top or something to get a feel for simple patterns. We are going to take it to the next level, and look at a quadratic pattern which will use systems of linear equations to find. These are one of my favorite parts of Algebra.
Easy Problem: Since Tuesday, we have been solving equations, we will keep that going. Tomorrow, we will look at a sequence, but today, we'll just keep it simple. In our equation, we will have two steps after you plug in b and simplify. First, you will have to get rid of some addition at the end by subtracting from both sides. Then, you will divide on both sides to reach your answer. Good luck!
Plug in b and solve for z: 100 = 6bz + 4
z = ___
Hard Problem: When given a sequence, it is essential that you create an equation that has you put in the spot that you are looking for and have it equal to the number in that spot. For instance, the sequence 3, 12, 21, 30, ... will have the equation n = 9x - 6 with x being the spot you put it in and n being the value in that spot. In order to figure out the equation, you need to look for common differences (for arithmetic sequences). In that one, you would have:
3 12 21 30
\/ \/ \/
9 9 9
In that case, we had the same differences at our first step. That means our equation is in the form y = mx+b. Hence, we create a system where we are solving for m and b by plugging in 1 for x and 3 for y in the first one. 2 for x and 12 for y is the second, and so on. You'll notice these equations are very easy to do elimination in because terms are already isolated.
If you don't get common differences, try finding the common differences of the common differences and see if those are the same. If they are, plug values into y = ax^2 + bx + c and solve for a, b, and c. This will require three equations.
1) Find the value for p and q.
s + 0.9/6 = p
t - 3.9 = q
2) Find the common differences in this sequence: p, 20, q, 48, ...
3) Create a system that could be used to find the equation for this sequence.
If you want to put yourself one step ahead, try to solve the system and determine what the equation is. Since we went over systems last month, you should be able to figure it out. Just remember to eliminate variables with no coefficient. Tomorrow, everybody is doing sequences! It's going to be fun!!