Section 9 2 Arithmetic Sequences and Partial Sums

Section 9. 2 Arithmetic Sequences and Partial Sums 1

Objective By following instructions students will be able to: § Recognize, write, and find the nth terms of arithmetic 2 sequences. § Find the nth partial sums of arithmetic sequences. § Use arithmetic sequences to model and solve real-life problems.

3 Definition of Arithmetic Sequence A sequence is arithmetic if the differences between consecutive terms are the same. So the sequence, is arithmetic if there is a number d such that The number d is the common difference of the arithmetic sequence.

Example 1: a) The sequence whose nth term is 4 n+3 is arithmetic. For this sequence the common difference between consecutive terms is 4. b) The sequence whose nth term is 7 -5 n is arithmetic. For this sequence, the common difference between consecutive terms -5. 4

Example 1: c) The sequence whose nth term is ¼(n+3) is arithmetic. For this sequence, the common difference between consecutive terms 1/4. d) The sequence 1, 4, 9, 16, …. , whose nth term is n 2 is not arithmetic. The difference between the first two terms is 3 and the difference between the second and third term is 5. 5

U-TRY #1 Determine whether the sequence is arithmetic. If it is, find the common difference. a) 4, 9, 14, 19, 24, …. . b) 1/3, 2/3, 4/3, 8/3, 16/3, …. c) 12, 22, 32, 42, 52, …. 6

7 The nth term of an Arithmetic Sequence The nth term of an arithmetic sequence has the form Where d is the common difference between consecutive terms of the sequence and .

Example 2: Find a formula for the nth term of the arithmetic sequence whose common difference is 3 and whose first term is 2. 8

Example 3: The fourth term of an arithmetic sequence is 20, and the 13 th term is 65. Write the first several terms of this sequence. 9

Example 4: Find the seventh term of the arithmetic sequence whose first two terms are 2 and 9. 10

U-TRY #2 1) The first two terms of the arithmetic sequence are given. Find the missing term. a) b) 2) Find a formula for an for the arithmetic sequence. 11

The Sum of a Finite Arithmetic Sequence The sum of a finite arithmetic series with n terms is 12

Example 5: Find the sum: 1+3+5+7+9+11+13+15+17+19 13

Example 6: Find the sum from 0 to 100. 14

Example 7: Find the 150 th partial sum of the arithmetic sequence 5, 16, 27, 38, 49, …. 15

Example 8: 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? 16

Example 9: A small business sells $10, 000 worth of products 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 this goal is met, find the total sales during the first 20 years this business is in operation. 17

U-TRY #3 1) Find the indicated nth partial sum of the arithmetic sequence. a) -6, -2, 2, 6, …. . , n=50 b) 2) Find the partial sum without using a graphing utility. 18

Revisit Objective Did we… § Recognize, write, and find the nth terms of arithmetic 19 sequences? § Find the nth partial sums of arithmetic sequence? § Use arithmetic sequences to model and solve real-life problems?

Homework § Pg 635 #s 1 -43 ODD, 20 59 -71 ODD, 85