Nth Term

Bellringer:


Homework:

How many triangles will there be in figure 7? Figure 57?


Classwork:

What is the formula for the top (using 'n')? What is the formula for the bottom (using 'n')?


Notes:

Today we discussed "nth" term rules. Basically we want to be able to write patterns in mathematical terms, instead of in sentences. We do this because it is much easier to use a formula to solve a pattern.

What I mean by solving a pattern is that sometimes you will be given a pattern, like 1, 2, 3, 4, ... and asked what the 40th number will be. In this case we can see that the first number is one, the second number is two, et cetera. Therefore, the 40th number will be 40. Mathematically the formula for this is:

n

"n" stands for the position a number falls into (i.e. First, second, third, ...), so when we plug in "1" for "n" because we want to know the first number we get: 1. Let's try a different one.

2, 3, 4, 5, ...

This pattern still increases by one each time, however, in order to get from our "n's" to our pattern we now have to add one. The formula for this is:

n + 1

Finally a more difficult pattern. 1, 4, 9, 16. Remember, n stands for the position of each number in the pattern. That means the the "n" for 1 is 1. The "n" for 4 is 2, the "n" for 9 is 3 and the "n" for 16 is 4. The question, again, is how can we turn the "n" into the pattern number. Let's try a few options.

The second number is 4, it's "n" is 2. Mathematically, what can we do to the number 2 to get it to become a 4. There are 3 options. The first is to add 2.:

2 + 2 = 4

This works for 2, but does it fit the pattern? If I add 2 to 3 (for the next part of the pattern) do I get 9?

3 + 2 = 5
3 + 2 ≠ 9

So, +2 does not give us the formula. The next option to get from 2 ("n") to 4 is to multiply by 2.

2 x 2 = 4

Does this work with 3?

2 x 3 = 6
2 x 3 ≠ 9

No. The last option is to multiply the "n" by itself.

2 x 2 = 4 or 2² = 4

It works for two, but will it work for 3?

3 x 3 = 9 or 3² = 9

Yes, it works for 3. It also works for the rest of the pattern. Therefore, the correct formula for this pattern is:

n²

Why bother finding nth terms? Many lines of work, including epidemiology (studying the spread of diseases) use formulas to describe complex patterns that occur in nature. This allows doctors to use patterns without having to write out entire sequences.