Arithmetic Sequences and Series

Sequences Series List with commas “Indicated sum” 3, 8, 13, 18 3 + 8 + 13 + 18

Arithmetic Sequences

An Arithmetic Sequence is defined as a sequence in which there is a common difference between consecutive terms.

Which of the following sequences are arithmetic Which of the following sequences are arithmetic? Identify the common difference. YES YES NO NO YES

The common difference is always the difference between any term and the term that proceeds that term. Common Difference = 5

The general form of an ARITHMETIC sequence. First Term: Second Term: Third Term: Fourth Term: Fifth Term: nth Term:

Formula for the nth term of an ARITHMETIC sequence. If we know any three of these we ought to be able to find the fourth.

Given: Find: IDENTIFY SOLVE

Find: What term number is -169? Given: Find: What term number is -169? IDENTIFY SOLVE

Given: Find: What’s the real question? The Difference IDENTIFY SOLVE

Given: Find: IDENTIFY SOLVE

Arithmetic Series

Write the first three terms and the last two terms of the following arithmetic series. What is the sum of this series?

What is the SUM of these terms? Written 1st to last. Written last to 1st. Add Down 50 Terms 71 + (-27) Each sum is the same.

In General . . .

Find the sum of the terms of this arithmetic series.

Find the sum of the terms of this arithmetic series. What term is -5?

Alternate formula for the sum of an Arithmetic Series.

Find the sum of this series It is not convenient to find the last term.

Your Turn

An introduction………… Arithmetic Series Sum of Terms Geometric Series Sum of Terms Arithmetic Sequences Geometric Sequences ADD To get next term MULTIPLY To get next term

Find the next four terms of –9, -2, 5, … Arithmetic Sequence 7 is referred to as the common difference (d) Common Difference (d) – what we ADD to get next term Next four terms……12, 19, 26, 33

Find the next four terms of 0, 7, 14, … Arithmetic Sequence, d = 7 21, 28, 35, 42 Find the next four terms of x, 2x, 3x, … Arithmetic Sequence, d = x 4x, 5x, 6x, 7x Find the next four terms of 5k, -k, -7k, … Arithmetic Sequence, d = -6k -13k, -19k, -25k, -32k

Vocabulary of Sequences (Universal)

Given an arithmetic sequence with x 38 15 NA -3 X = 80

-19 353 ?? 63 x 6

Try this one: 1.5 16 x NA 0.5

9 x 633 NA 24 X = 27

-6 29 20 NA x

Find two arithmetic means between –4 and 5 -4, ____, ____, 5 -4 5 4 NA x The two arithmetic means are –1 and 2, since –4, -1, 2, 5 forms an arithmetic sequence

Find three arithmetic means between 1 and 4 1, ____, ____, ____, 4 1 4 5 NA x The three arithmetic means are 7/4, 10/4, and 13/4 since 1, 7/4, 10/4, 13/4, 4 forms an arithmetic sequence

Find n for the series in which 5 y x 440 3 Graph on positive window X = 16

Example: The nth Partial Sum The sum of the first n terms of an infinite sequence is called the nth partial sum. Example: The nth Partial Sum

Example 6. Find the 150th partial sum of the arithmetic sequence, 5, 16, 27, 38, 49, …

Example 7. An auditorium has 20 rows of seats Example 7. An auditorium has 20 rows of seats. There are 20 seats in the first row, 21 seats in the second row, 22 seats in the third row, and so on. How many seats are there in all 20 rows?

Example 8. A small business sells $10,000 worth of sports memorabilia during its first year. The owner of the business has set a goal of increasing annual sales by $7500 each year for 19 years. Assuming that the goal is met, find the total sales during the first 20 years this business is in operation. So the total sales for the first 2o years is $1,625,000