1. 21. Sequences
A list of numbers following a certain pattern
a1 , a2 , a3 , a4 , … , an , …
Pattern is determined by position or by what has come before
3, 6, 12, , , …

2. Defined by n(position)
Find the first four terms and the 100th term for the following:

3. Defined Recursively
Find the first five terms for the following:

4. Partial Sums
Adding the first n terms of a sequence, the nth partial sum:

5. Partial Sums – continued
Find the first 4 partial sums and then the
nth partial sum for the sequence defined by:

6. Sigma Notation,  Write the following without sigma notation:
Find the value of the sum:

7. 22. Arithmetic Sequences 4, 7, 10, 13, 16, … 81, 75, 69, 63, 57, …
Consider the following sequences:
4, 7, 10, 13, 16, …
81, 75, 69, 63, 57, …

8. Definition An arithmetic sequence is the following:
with a as the first term and d as the common difference.

9. Example 1

10. Partial Sum
Find the 1000th partial sum for arithmetic sequence with a = 1, d = 1:

11. Partial Sum – continued
Find the 7th partial sum for arithmetic sequence with a = 10, d = 7:

12. Example 2
For the month of June, you got $4 on the first day, $7 on the second day, $10 on the third day and so on. How much did you receive in total by the end of the month?

13. 23. Geometric Sequences Consider the following sequences:
3, 6, 12, 24, 48, …
81, 27, 9, 3, 1, …

14. Definition A geometric sequence is the following:
with a as the first term and r as the common ratio.

15. Examples
Find the common ratio, the nth term, and the 5th term.

16. Partial Sum & Infinite Sums
The partial sum of a geometric sequence looks like:

17. Infinite Sums

18. Examples
Find the sum of the infinite geometric series:

19. 24. Mathematics of Finance
Two situations that use geometric sequences:
Annuity – money paid in equal payments to become a lump sum (the amount of the annuity, Af, is the sum of all payments plus any interest accrued.)
Installment Buying – lump sum borrowed now and paid back in equal payments,
how much is paid monthly on a loan for a car, house, …?
how much can be borrowed if a certain monthly payment is possible?

20. Annuities
Make yearly payments of $200 to an account that has an annual interest rate of 5%. What is the account worth after the 10th payment is made?

21. Annuities (in general)

22. Installment Buying

23. Example 1

24. Example 2