1. Arithmetic Sequences

2. 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,
is an example of an arithmetic sequence.

3. 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.

4. Text Example
The recursion formula an = an models the thousands of Air Force personnel on active duty for each year starting with 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
Solution The recursion formula an = an 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.

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

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

7. 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

8. 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.

9. 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.

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

11. Example
Find the indicated sum
Solution:

12. Arithmetic Sequences