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, 24, 48, …

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

Defined Recursively Find the first five terms for the following:

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

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

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

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, …

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

Example 1

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

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

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?

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

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

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

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

Infinite Sums

Examples Find the sum of the infinite geometric series:

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?

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?

Annuities (in general)

Installment Buying

Example 1

Example 2