Objectives (click/tap to expand)
Find specific terms of explicitly-defined sequences
Find specific terms of recursively-defined sequences
Determine whether a sequence converges or diverges
Analyze and define sequences
Graph sequences
Use summation notation (a.k.a. sigma notation)
Find the sums of arithmetic and geometric sequences
Find the sum if an infinite geometric sequence converges
Express a rational number as a fraction of integers
Solve application problem
Yes, at first glance this appears a random thing to throw in at the end of the course.
However, this is essential material in preparation for Calculus. Interestingly, there are many strong connections between this section and Chapter 3, so the sequencing of course material actually works quite well.
Note: In the InteractMath site, this is listed as both Chapters 9.4 and 9.5. This is because we use a 6th edition textbook but select 7th edition in InteractMath. This section was split into two sections between the editions.
Introduction to Sequences, which are an essential bridge to Calculus
When given just two specific terms of a sequence, construct a recursive and explicit rule for the sequence. What other piece of information do you need in order to do this?
SEE IF YOU CAN RELATE THIS TO THE CHAPTER 3.1 LESSON in which we construct an exponential function that passes through two given points. Conceptually, these two lessons identical.
Writing arithmetic series in sigma notation (a.k.a. summation notation) and then calculating the value of the sum
Writing geometric series in sigma notation (a.k.a. summation notation) and then calculating the value of the sum. This topic is covered in class, but check out the video if you'd like to see/hear the explanation again. The examples covered in this video are as challenging as anything I'll throw at you on an assignment or test for this chapter.
The "long way" and "short way" of computing the sum of converging infinite geometric series.
If you would like more practice on recent material, here are some Sequence/Series exercises & answers from two other textbooks (PreCalculus by Larson & Hostetler, 7th edition, and PreCalculus by Sullivan, 4th edition).
This assessment covers Chapters 3.4, 3.5, and 9.4 with topics as follows:
Chapter 3.4
Use logarithmic properties to "expand" a logarithm into a sum/difference of logs and/or multiples of logs
Use logarithmic properties to combine multiple log expressions into a single log
Use change of base formula to write log expressions using only common logs or natural logs
Chapter 3.5
Solve exponential equations (a) algebraically, using same base and one-to-one properties, (b) algebraically, using logarithms, (c) graphically
Solve logarithmic equations (a) algebraically, (b) graphically
Reminder that there were extra exercises (with solutions) used as classwork several lessons ago
Exponential/logarithmic exercises from another textbook, with odd answers (Larson, 7th ed)
More exercises (with answers) made by Mr. W, similar to our textbook
Sketching/labeling a graph showing how an object's temperature cools, and developing a T(t) equation for temperature as a function of time (a.k.a. Newton's Law of Cooling).
Check out the additional practice for Newton's Law of Cooling (solutions included).
Chapter 9.4
Given an explicit or recursive rule, write out the terms of a sequence
Given the terms of an arithmetic or geometric sequence, write an explicit or recursive rule
Construct an arithmetic or geometric sequence, given any two terms of the sequence
Write a series in sigma notation (a.k.a. summation notation)
Calculate the sum of a finite arithmetic or geometric sequence
Determine whether an infinite series converges or diverges IGNORE THIS FOR 2019
Calculate the sum of a converging infinite series IGNORE THIS FOR 2019
"Ugly Algebra"
This "Ugly Algebra" PDF includes eight worked-out examples and eight exercises for you to try on your own, with answers on the last pages.
Clever video. Do you see the references to infinite geometric series and other recursive operations?
Another glimpse into the struggle to wrap your head around the confusion over infinity.
Go to next page, Introduction to Calculus.