1. Section 11.1 Sequences

2. The following are all examples of sequences (do you know what kind of sequences they are?)
0, 1, 4, 9, 16, 25, 36,…
1, 1, 2, 3, 5, 8, 13, 21, 34, …
2, 4, 6, 8, 10, 12, 14, 16,…
4, 12, 36, 108, 324, 972,…
80, 82, 76, 75, 100, 130, 150,…

3. 0, 1, 4, 9, 16, 25, 36,…sequence of squares of integers
1, 1, 2, 3, 5, 8, 13, 21, 34, …Fibonacci sequence
2, 4, 6, 8, 10, 12, 14, 16,…sequence of even numbers
4, 12, 36, 108, 324, 972,…sequence of numbers multiplied by 3 beginning with 4
80, 82, 76, 75, 100, 130, 150,…my monthly electric bill from SRP beginning in January

4. Notation for sequences
We denote the terms of the sequence by
Where a1 is the first term, a2 is the second term, and so on
We use an to denote the nth or general term of the sequence
If there is a pattern to the sequence we may be able to find a formula for an

5. Consider the sequence 2, 6, 10, 14, 18, 22,…
Do you notice any pattern?
Any sequence that has a common difference between terms is called an arithmetic sequence
Let’s see if we can find the formula for the nth term of the sequence
What if the first term was 3 instead of 2 (still with a common difference of 4)?
From here we can see the general formula of an arbitrary arithmetic sequence with a common difference, d

6. The general formula of an arbitrary arithmetic sequence with a common difference, d
Notice that this is a linear function with a slope of d
A sequence is a function whose domain is the set of Natural Numbers (e.g., 1, 2, 3, 4, 5, 6,…)
We can take the function f(x) = 3x – 2 and write an arithmetic sequence with f(1), f(2), f(3), and so on
The main difference is we use n as the independent variable and use subscript notation instead of function notation

7. Consider the sequence 3, 6, 12, 24, 48, 96,…
Do you notice any pattern?
Any sequence that has a common ratio (or common multiple) between consecutive terms is called an geometric sequence
Let’s see if we can find the formula for the nth term of the sequence
From here we can see the general formula of an arbitrary geometric sequence with a common ratio, r

8. From here we can see the general formula of an arbitrary geometric sequence with a common ratio, r
Notice that we have an exponential function with an input variable of n and a base of r

9. For the following sequences, identify which are arithmetic, which are geometric, and which are neither. Give the 100th term for each.