A list of numbers following a certain pattern a 1, a 2, a 3, a 4, …, a n, … Pattern is determined by position or by what has come before 25. Sequences 3, 6, 12, 24, 48, … 1

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

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

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

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

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

Consider the following sequences: 26. Arithmetic Sequences 4, 7, 10, 13, 16, … 81, 75, 69, 63, 57, … 7

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

Example 1 9

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

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

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? 12

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

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

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

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

Infinite Sums 17

Examples Find the sum of the infinite geometric series: 18

Two situations that use geometric sequences:  Annuity – money paid in equal payments to become a lump sum (the amount of the annuity, A f, 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? 28. Mathematics of Finance 19

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

Annuities (in general) 21

Installment Buying 22

Example 1 23

Example 2 24