College Algebra 9th txtbk

522 CHAPTER 8 SEQUENCES, INDUCTION, AND PROBABILITY
Z Developing nth-Term Formulas
If {a n } is an arithmetic sequence with common difference d, then
a 2 a 1 d
a 3 a 2 d a 1 2d
a 4 a 3 d a 1 3d
This suggests Theorem 1, which can be proved by mathematical induction (see Problem 67
in Exercises 8-3).
Z THEOREM 1 The nth Term of an Arithmetic Sequence
a n a 1 (n 1)d for every n 1
Similarly, if {a n } is a geometric sequence with common ratio r, then
a 2 a 1 r
a 3 a 2 r a 1 r 2
a 4 a 3 r a 1 r 3
This suggests Theorem 2, which can also be proved by mathematical induction (see Problem
71 in Exercises 8-3).
Z THEOREM 2 The nth Term of a Geometric Sequence
a n a 1 r n1 for every n 1
EXAMPLE 2 Finding Terms in Arithmetic and Geometric Sequences
(A) If the first and tenth terms of an arithmetic sequence are 3 and 30, respectively, find
the fiftieth term of the sequence.
(B) If the first and tenth terms of a geometric sequence are 1 and 4, find the seventeenth
term to three decimal places.
SOLUTIONS (A) First use Theorem 1 with a 1 3 and a 10 30 to find d:
a n a 1 (n 1)d
a 10 a 1 (10 1)d
30 3 9d
d 3
Substitute n 10.
Substitute a 10 30 and a 1 3.
Solve for d.
Now find a 50 :
a 50 a 1 (50 1)3
3 49 3
150
Substitute a 1 3.
Simplify.
(B) First let n 10, a 1 1, a 10 4 and use Theorem 2 to find r.
a n a 1 r n1 Substitute n 10, a 10 4, and a 1 1.
4 1r 101 Solve for r.
r 4 19