Arithmetic Sequences and Partial Sums A sequence whose consecutive terms have a common difference is called an arithmetic sequence. A sequence is arithmetic if the differences between consecutive terms are the same. 4, 9, 14, 19, 24, . . . arithmetic sequence 9 – 4 = 5 14 – 9 = 5 19 – 14 = 5 24 – 19 = 5 The common difference, d, is 5. Definition of Arithmetic Sequence

Example 1. Are the following arithmetic sequences?

The nth Term of an Arithmetic Sequence The nth term of an arithmetic sequence has the form an = dn + c where d is the common difference and c = a1 – d. a1 = 2 c = 2 – 6 = – 4 2, 8, 14, 20, 26, . . . . d = 8 – 2 = 6 The nth term is 6n – 4. The nth Term of an Arithmetic Sequence

Let’s look at the following sequence. 2, 6, 10, 14, 18, … d = 4

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

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

If you know the nth term of an arithmetic sequence and you know the common difference of the sequence, you can find the (n + 1)th term by using the recursive formula. With this formula, you can find any term of an arithmetic sequence if you know the preceding term.

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

The Sum of a Finite Arithmetic Sequence The sum of a finite arithmetic sequence with n terms is given by 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 = ? n = 10 a1 = 5 a10 = 50 The Sum of a Finite Arithmetic Sequence

Example 5. Find each sum. 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 Sum of the integers from 1 to 100

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