9 2 ARITHMETIC SEQUENCES AND PARTIAL SUMS Copyright

9. 2 ARITHMETIC SEQUENCES AND PARTIAL SUMS Copyright © Cengage Learning. All rights reserved.

What You Should Learn • Recognize, write, and find the nth terms of arithmetic sequences. • Find n th partial sums of arithmetic sequences. • Use arithmetic sequences to model and solve real-life problems. 2

Arithmetic Sequences A sequence whose consecutive terms have a common difference is called an arithmetic sequence. 3

Example 1 – Examples of Arithmetic Sequences For each of the arithmetic sequences, find the common difference. a. 5, 8, 11, 14, 17, … 3 n + 2 b. 9, 5, 1, -3, -7, …, 13 – 4 n c. 4

Arithmetic Sequences The sequence 1, 4, 9, 16, . . . , whose n th term is n 2, is not arithmetic. The difference between the first two terms is a 2 – a 1 = 4 – 1 = 3 but the difference between the second and third terms is a 3 – a 2 = 9 – 4 = 5. 5

Example 2 – Finding the nth Term of an Arithmetic Sequence Find a formula for the n th term of the arithmetic sequence whose common difference is 3 and whose first term is 2. 6

Example 3 The eight term of an arithmetic sequence is 25, and the 12 th term is 41. Write the first 5 terms of this sequence. 7

Arithmetic Sequences If you know the n th 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 recursion formula an + 1 = an + d. Recursion formula With this formula, you can find any term of an arithmetic sequence, provided that you know the preceding term. For instance, if you know the first term, you can find the second term. Then, knowing the second term, you can find the third term, and so on. 8

Example 4 Find the 10 th term of the arithmetic sequence that begins with 8 and 20. 9

The Sum of a Finite Arithmetic Sequence There is a simple formula for the sum of a finite arithmetic sequence. 10

Example 5 – Finding the Sum of a Finite Arithmetic Sequence Find the sums of: a. 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19. b. Integers from 1 to 35 c. Integers from 1 to 57 11

The Sum of a Finite Arithmetic Sequence The sum of the first n terms of an infinite sequence is the n th partial sum. The n th partial sum can be found by using the formula for the sum of a finite arithmetic sequence. 12

Example 6 Find the 50 th partial sum of the arithmetic sequence, -6, -2, 2, 6, …. 13

Example 8 – Prize Money In a golf tournament, the 16 golfers with the lowest scores win cash prizes. First place receives a cash prize of $1000, second place receives $950, third place receives $900, and so on. What is the total amount of prize money? 14