5.7 Arithmetic & Geometric Sequences
An arithmetic sequence (sometimes called an arithmetic progression) is one in which the difference between
successive terms of the sequence is always the same number. It can be defined recursively as a1 = a, an = an-1 +
d, where a and d are real numbers. The number a is the first term and d is the common difference. A sequence
can be shown to be arithmetic if subtracting successive terms yields a constant. For an arithmetic sequence {an}
whose first term is a and whose common difference is d, the nth term or general term is determined by the
formula an = a + (n – 1)d.
Determine if the following sequences are arithmetic:
1. 5, 9, 13, 17, …
2.
Find the 50th term of 5, 9, 13, 17, …
3.
Find the 20th term of {sn} = {5n + 9}
Find the general formula for the nth term of the arithmetic sequence; then find the 18th term:
5. 4 + 7 + 10 + …
6.
Suppose that you take a job that starts at $25,000 and promises a $500 increase every year.
a. Write a general formula for the nth term of this arithmetic sequence.
b.
What will you be earning in the year 2020?
An geometric sequence (sometimes called an geometric progression) is one in which the ratio of
successive terms of the sequence is always the same nonzero number. It can be defined recursively
as a1 = a, an = r(an-1) or an/ an-1 = r, n > 0, where a and r ≠ 0 are real numbers. The number a is the first
term and r is the common ratio. A sequence can be shown to be geometric if dividing successive
terms yields a constant. For a geometric sequence {an} whose first term is a and whose common ratio
is r, the nth term is determined by the formula an = a1rn-1, r ≠ 0.
Determine if the following sequences are arithmetic or geometric:
7. 2, 6, 8, 10 …
 
n
8.
{sn} = 3
9.
1 1
Find the 7th term of 1, , ,...
2 4
2
10. Find the 20th term of the geometric sequence in which a = 7 and r = 3
Write the general term for the following and find the 32th term:
11. 1, -5, 25, -125, …
12. Suppose that you take a job that starts at $25,000 and promises a 2% increase every year.
a. Write a general formula for the nth term of this geometric sequence.
b.
What will you be earning in the year 2020?
13. Problem 116, page 261
a.
b.
14. Problem 124, page 262
a.
b.
c.