Copyright © Cengage Learning. All rights reserved. 9 Sequences, Series, and Probability Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved. 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.

Arithmetic Sequences

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

Example 1 – Examples of Arithmetic Sequences a. The sequence whose n th term is 4n + 3 is arithmetic. For this sequence, the common difference between consecutive terms is 4. 7, 11, 15, 19, . . . , 4n + 3, . . . b. The sequence whose nth term is 7 – 5n is arithmetic. For this sequence, the common difference between consecutive terms is – 5. 2, –3, – 8, –13, . . . , 7 – 5n, . . . Begin with n = 1. 11 – 7 = 4 Begin with n = 1. –3 – 2 = –5

Example 1 – Examples of Arithmetic Sequences cont’d c. The sequence whose nth term is is arithmetic. For this sequence, the common difference between consecutive terms is Begin with n = 1.

Arithmetic Sequences The sequence 1, 4, 9, 16, . . . , whose n th term is n2, is not arithmetic. The difference between the first two terms is a2 – a1 = 4 – 1 = 3 but the difference between the second and third terms is a3 – a2 = 9 – 4 = 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. Solution: You know that the formula for the n th term is of the form an = a1 + ( n – 1)d. Moreover, because the common difference is d = 3 and the first term is a1 = 2, the formula must have the form an = 2 + 3(n – 1). Substitute 2 for a 1 and 3 for d.

Example 2 – Solution So, the formula for the n th term is an = 3n – 1. cont’d So, the formula for the n th term is an = 3n – 1. The sequence therefore has the following form. 2, 5, 8, 11, 14, . . . , 3n – 1, . . .

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. 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. Recursion formula

The Sum of a Finite Arithmetic Sequence

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

Example 5 – Finding the Sum of a Finite Arithmetic Sequence Find the sum: 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19. Solution: To begin, notice that the sequence is arithmetic (with a common difference of 2). Moreover, the sequence has 10 terms. So, the sum of the sequence is Sn = (a1 + an) Formula for the sum of an arithmetic sequence

Example 5 – Solution = (1 + 19) = 5(20) = 100. cont’d = (1 + 19) = 5(20) = 100. Substitute 10 for n, 1 for a1, and 19 for an. Simplify.

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.

Applications

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? Solution: The cash prizes awarded form an arithmetic sequence in which the first term is a1 = 1000 and the common difference is d = – 50.

Example 8 – Solution Because an = 1000 + (– 50)(n – 1) cont’d Because an = 1000 + (– 50)(n – 1) you can determine that the formula for the n th term of the sequence is an = – 50n + 1050. So, the 16th term of the sequence is a16 = – 50(16) + 1050 and the total amount of prize money is S16 = 1000 + 950 + 900 + . . . + 250 = 250,

Example 8 – Solution S16 = (a1 + a16) = (1000 + 250) = 8(1250) cont’d S16 = (a1 + a16) = (1000 + 250) = 8(1250) = $10,000. n th partial sum formula Substitute 16 for n, 1000 for a1, and 250 for a16. Simplify.