COLLEGE ALGEBRA
These brief notes are intended to guide you through the textbook and/or other course readings/materials. As you read the textbook pay particular attention to the "topics of importance" and be sure you know how to accomplish each. The "supplemental sites" may provide additional resources on the internet that supplement the topics. Note: this material is extensively elaborated upon in my optional e-book GOLDen Mathematics: College Algebra. This section of material only is downloadable for a nominal fee at <<T.B.A.>>. See "Tell me more about Keely's GM book".
Introduction to Sequences and Series
GOLDen Mathematics - College Algebra: Section 7.1
Supplemental Sites:
MathOL Links - Calg 7.1
Topics of Importance
terminology associated with sequences, series, and sums
Factorials: notation, evaluate, simplify
Sequences: notation, formula 
terms
Recursively defined sequences
Summations: notation, evaluate (by hand and on calc), simplify
Series: formula
terms
Properties of series
Comments and Cautions
Hello everyone. Are you ready? Pop quiz! Can you complete
these sequences?
a. 5, 11, 17, 23, 29, ___, ___, ___
b. 2, 4, 8, 16, 32, ___, ___, ___
c. 3, 5, 7, 11, 13, ___, ___, ___
d. 1, 1, 2, 3, 5, 8, ___, ___, ___
Think about them before reading on.
Answers:
a. 35, 41, 47. To get the subsequent terms you add 6. This is an example of an
"arithmetic sequence".
b. 64, 128, 256. To get the subsequent terms you multiply by 2. This is an
example of a "geometric sequence".
c. 17, 19, 23. This is simply a sequence of prime numbers.
d. 13, 21, 34. Can you see that the subsequent terms are formed by combining
previous terms (1+1=2, 1+2=3, 2+3=5, 3+5=8, etc.)? This is a special sequence
which is an example of a "recursive sequence".
This last chapter of material in the College Algebra course involves sequences such as these. This lesson introduces general sequences (lists) and series (sums) -- the notation, terminology, and processes. We'll follow this lesson up with a concentration on specific types of sequences and series (arithmetic and geometric). This material will be used particularly in Calculus III when we will write functions such as ex as a sum of infinitely many rational terms (ex is equal to 1 + x + x2/2! + x3/3! + x4/4! + ...). This will enable us to perform calculus on rational expressions rather than on the (often more complicated) transcendental function itself.
My online Calculator Guide: Sequences and Series is available to help you evaluate sequences and series on your graphing calculator. You should be able to do so both using a calculator and algebraically by hand.
Text Notes (These notes refer to College Algebra 4th ed by Blitzer section 8.1.)
 | ch 8.1 begins by introducing the Fibonacci Sequence, the most famous example of a recursive sequence. If you haven't investigated this very rich topic before or would like to learn more, please see my MathOL Links - Liberal Arts Math - Fibonacci Sequence. |
| ch 8.1 introduces lots of terminology for sequences, series, and sums. These terms will be used throughout the chapter as we move from general sequences/series to very specific sequences/series. Several method of recognizing patterns are also discussed. |
Arithmetic Sequences and Series
GOLDen Mathematics - College Algebra: Section 7.2
Supplemental Sites:
MathOL Links - Calg 7.2
Topics of Importance
Arithmetic sequences: definition, n-th term formula,
find specific term, find number of terms
Arithmetic series: definition, sum formula, find specific term, find number of
terms, write in summation notation
Applications of arithmetic sequences and series
Comments and Cautions
Our studies today take us into the first of two special sequences.
Sequences such as 4, 11, 18, 25, 32, ... where you add (or
subtract) a fixed number to get the next terms are arithmetic. This particular
sequence can be written in general as {4+7(n-1)}={7n-3}. It is important to be
able to write the first few terms given this general formula or derive it when
given the sequence's terms. Arithmetic series are sums where the terms are
arithmetic such as 4+11+18+25+32+.... Something to think about: do all infinite
arithmetic series have infinite sums?
Text Notes (These notes refer to College Algebra 4th ed by Blitzer section 8.2.)
| ch 8.2 Concentrate on the algebraic techniques, particularly finding the n-th term, number of terms, and sum. |
| ch 8.2 It is not important to be able to produce the graphs such as the one shown on page 659. |
Geometric Sequences and Series
GOLDen Mathematics - College Algebra: Section 7.3
Supplemental Sites:
MathOL Links - Calg 7.3
Topics of Importance
Geometric sequences: definition, n-th term formula, find
specific term, find number of terms
Geometric series: definition, sum formulas (for both finite and infinite
series),
find specific term, find number of terms, write in summation notation
Converting repeating decimals to fractions
Applications of geometric sequences and series
Comments and Cautions
Our studies today take us into the second of two special sequences.
Sequences such as 5, 10, 20, 40, 80, ... where you multiply
a fixed number to get the next terms are geometric. This particular sequence can
be written in general as {5*2n-1}. Again it is important to be able
to write the first few terms given this general formula or derive it when given
the sequence's terms. Geometric series are sums where the terms are geometric
such as 5+10+20+40+80+.... Something to think about: Some infinite geometric
series have finite sums. Under what conditions do infinite geometric series have
finite sums?
Text Notes (These notes refer to College Algebra 4th ed by Blitzer section 8.3.)
| ch 8.3 Concentrate on the algebraic techniques, particularly finding the n-th term, number of terms, and sum. |
| ch 8.3 Pay attention to the differences between a finite and infinite geometric sum. |
| ch 8.3 example 9 shows a rather tedious method of converting a repeating decimal to a fraction. It may be somewhat cool, but simple algebra can be used to accomplish the same conversion. If you want to see the steps, ask in class and we can review the elementary algebra method of conversion. |
| ch 8.4-8.7 SKIP. |
| And on that note we conclude the course material, woohoo!! |
Originally written: 2006-09-04
Last revision:
2008-06-02 02:17 AM