Arithmetic Sequences

Arithmetic Sequences A mathematical model for the average annual salaries of major league baseball players generates the following data. 1,438,000 1,347,000 1,256,000 1,165,000 1,074,000 983,000 892,000 801,000 Salary 1998 1997 1996 1995 1994 1993 1992 1991 Year From 1991 to 1992, salaries increased by $892,000 - $801,000 = $91,000. From 1992 to 1993, salaries increased by $983,000 - $892,000 = $91,000. If we make these computations for each year, we find that the yearly salary increase is $91,000. The sequence of annual salaries shows that each term after the first, 801,000, differs from the preceding term by a constant amount, namely 91,000. The sequence of annual salaries 801,000, 892,000, 983,000, 1,074,000, 1,165,000, 1,256,000.... is an example of an arithmetic sequence.

Definition of an Arithmetic Sequence An arithmetic sequence is a sequence in which each term after the first differs from the preceding term by a constant amount. The difference between consecutive terms is called the common difference of the sequence.

Text Example The recursion formula an = an - 1 - 24 models the thousands of Air Force personnel on active duty for each year starting with 1986. In 1986, there were 624 thousand personnel on active duty. Find the first five terms of the arithmetic sequence in which a1 = 624 and an = an - 1 - 24. Solution The recursion formula an = an - 1 - 24 indicates that each term after the first is obtained by adding -24 to the previous term. Thus, each year there are 24 thousand fewer personnel on active duty in the Air Force than in the previous year.

Text Example cont. The recursion formula an = an - 1 - 24 models the thousands of Air Force personnel on active duty for each year starting with 1986. In 1986, there were 624 thousand personnel on active duty. Find the first five terms of the arithmetic sequence in which a1 = 624 and an = an - 1 - 24. Solution a1 = 624 This is given. a2 = a1 – 24 = 624 – 24 = 600 Use an = an - 1 - 24 with n = 2. a3 = a2 – 24 = 600 – 24 = 576 Use an = an - 1 - 24 with n = 3. a4 = a3 – 24 = 576 – 24 = 552 Use an = an - 1 - 24 with n = 4. a5 = a4 – 24 = 552 – 24 = 528 Use an = an - 1 - 24 with n = 5. The first five terms are 624, 600, 576, 552, and 528.

Example Write the first six terms of the arithmetic sequence where a1 = 50 and d = 22 Solution: a1 = 50 a2 = 72 a3 = 94 a4 = 116 a5 = 138 a6 = 160

General Term of an Arithmetic Sequence The nth term (the general term) of an arithmetic sequence with first term a1 and common difference d is an = a1 + (n-1)d

Text Example Find the eighth term of the arithmetic sequence whose first term is 4 and whose common difference is -7. Solution To find the eighth term, as, we replace n in the formula with 8, a1 with 4, and d with -7. an = a1 + (n - 1)d a8 = 4 + (8 - 1)(-7) = 4 + 7(-7) = 4 + (-49) = -45 The eighth term is -45. We can check this result by writing the first eight terms of the sequence: 4, -3, -10, -17, -24, -31, -38, -45.

The Sum of the First n Terms of an Arithmetic Sequence The sum, Sn, of the first n terms of an arithmetic sequence is given by in which a1 is the first term and an is the nth term.

Example Find the sum of the first 20 terms of the arithmetic sequence: 6, 9, 12, 15, ... Solution:

Example Find the indicated sum Solution:

Arithmetic Sequences