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Lesson 24: The geometric series

Hi everyone! Read through the material below, watch the videos.

Lesson 24: The geometric series

Topic: This lesson covers Chapter 24: The geometric series

WeBWorK: There are two WeBWorK assignments on today’s material:

Sequences – Geometric

Series – Geometric

Question of the day: Can we add up infinitely many numbers?

Lesson Notes (Notability – pdf):

This .pdf file contains most of the work from the videos in this lesson. It is provided for your reference.

Finite geometric series

Today we look at a new kind of sequence, called a geometric sequence, and the corresponding series, geometric series.

A geometric sequence is a sequence for which we multiply by a constant number to get from one term to the next, for example:

An example of a geometric sequence

Definition 24.1. A sequence \left\{a_{n}\right\} is called a geometric sequence, if any two consecutive terms have a common ratio r. The geometric sequence is determined by r and the first value a_{1}. This can be written recursively as:

    \[a_{n}=a_{n-1} \cdot r \quad \text { for } n \geq 2\]

Alternatively, we have the general formula for the n th term of the geometric sequence:

    \[a_{n}=a_{1} \cdot r^{n-1}\]

Example 24.2. Determine if the sequence is a geometric or arithmetic sequence, or neither or both. If it is a geometric or arithmetic sequence, then find the general formula.
a) 3,6,12,24,48, \dots
b) 100,50,25,12.5, \ldots
c) 700,-70,7,-0.7,0.07, \ldots
d) 2,4,16,256, \dots
e) 3,10,17,24, \ldots
f) \quad-3,-3,-3,-3,-3, \dots
g) a_{n}=\left(\frac{3}{7}\right)^{n}
h) a_{n}=n^{2}

VIDEO: Introduction to geometric sequences, Example

Example 24.3. Find the general formula of a geometric sequence with the given property
a) r=4, and a_{5}=6400
b) a_{1}=\frac{2}{5}, and a_{4}=-\frac{27}{20}
c) a_{5}=216, a_{7}=24, and r is positive

VIDEO: Finding the formula of a geometric sequence – Example 24.3

Example 24.4. Consider the geometric sequence a_{n}=8 \cdot 5^{n-1}, that is the sequence:

    \[8,40,200,1000,5000,25000,125000, \ldots\]

Find the sum of the first 6 terms of this sequence

    \[8+40+200+1000+5000+25000=31248\]

VIDEO: Sum of a geometric series – intro example

Observation 24.5. Let \left\{a_{n}\right\} be a geometric sequence, whose n th term is given by the formula a_{n}=a_{1} \cdot r^{n-1} . We furthermore assume that r \neq 1 . Then, the sum a_{1}+a_{2}+\dots+a_{k-1}+a_{k} is given by

    \[\sum_{i=1}^{k} a_{i}=a_{1} \cdot \frac{1-r^{k}}{1-r}\]

Example 24.6. Find the value of the geometric series.
a) Find the sum \sum_{n=1}^{6} a_{n} for the geometric sequence a_{n}=10 \cdot 3^{n-1}
b) Determine the value of the geometric series: \sum_{k=1}^{5}\left(-\frac{1}{2}\right)^{k}
c) Find the sum of the first 12 terms of the geometric sequence

    \[-3,-6,-12,-24, \dots\]

VIDEO: The sum of a finite geometric series, Example 24.6

Infinite geometric series

Sometimes it makes sense to add up not just a finite number of terms in a sequence, but ALL the terms (infinitely many!).

Example 24.7. Consider the geometric sequence

    \[1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots\]


What is the initial term? What is the common ratio?
Let’s try adding up some of the terms. Try this by hand, and by using the formula for finite geometric series. What happens if we add up ALL the terms?

VIDEO: Infinite geometric series – intro example and formula

Definition 24.8. An infinite series is given by the formula

    \[\sum_{i=1}^{\infty} a_{i}=a_{1}+a_{2}+a_{3}+\ldots\]

Observation 24.9. Let \left{a_{n}\right} be a geometric sequence with a_{n}=a_{1} \cdot r^{n-1} Then the infinite geometric series is defined whenever -1<r<1. In this case, we have:

    \[\sum_{i=1}^{\infty} a_{i}=a_{1} \cdot \frac{1}{1-r}\]

\quad

Example 24.10. Find the value of the infinite geometric series.
a) \sum_{j=1}^{\infty} a_{j}, for a_{j}=5 \cdot\left(\frac{1}{3}\right)^{j-1}
b) \sum_{n=1}^{\infty} 3 \cdot(0.71)^{n}
c) 500-100+20-4+\ldots
d) 3+6+12+24+48+\ldots

Example 24.11. Consider the real number given by 0.555555\dots. Rewrite this number as an infinite geometric series. Can you figure out what fraction it is equal to?

VIDEO: Infinite geometric series – examples

Lesson 13: Exponential and Logarithmic Functions

Hi everyone! Read through the material below, watch the videos, and send me your questions.

Lesson 13: Exponential and Logarithmic Functions

Topic: This lesson covers Chapter 13 in the book, Exponential and Logarithmic Functions.

WeBWorK: There are two WeBWorK assignments on today’s material: Exponential Functions - Graphs and Logarithmic Functions - Graphs .

Lesson Notes (Notability – pdf):

This .pdf file contains most of the work from the videos in this lesson. It is provided for your reference.

Exponential Functions and their Graphs

We’ve been living in the world of Polynomials and Rational Functions. We now turn to exponential functions. These functions are “very natural” – that is, they show up in the real world – but they are also more complicated than Polynomial and Rational functions (for example, an exponential function grows more quickly than any Polynomial)

The spread of coronavirus, like other infectious diseases, can be modeled by exponential functions.

Definition. An exponential function is a function of the form f(x)=c\cdot b^x, where b and c are real numbers and b is positive (b is called the base, x is the exponent).

Example 1 (Textbook 13.2): Graph the exponential functions f(x)=2^x, g(x)=3^x, h(x)=10^x, k(x)=\left(\frac{1}{2}\right)^x, l(x)=\left(\frac{1}{10}\right)^x.

Now let’s see what happens when we change the number c in y=c\cdot b^x.

Example 2 (Textbook 13.6): Graph the exponential functions
a) y=2^{x}, \quad b) y=3 \cdot 2^{x}, \quad c) y=(-3) \cdot 2^{x}, \quad d) y=0.2 \cdot 2^{x}, \quad e) y=(-0.2) \cdot 2^{x}

Example 3: The graph below shows an exponential function f(x). Find a formula for f(x).

Logarithmic Functions and their Graphs

Definition. If b is a positive real number and b\neq 1, then the logarithm with base b is defined:
y=\log_b(x) \iff b^y=x

What does the definition of logarithm mean? The idea is that the logarithm is the inverse function of the exponential function. Let’s look at an example.

Question: Is an exponential function one-to-one? (What does one-to-one means).

Example 4. The graph below shows the function y=\log_2(x) but shifted to the right 3 units. Find a formula for the function in the graph.

That’s it for now! Take a look at the WeBWorK assignment, leave your questions below (or use the Ask for Help button in WeBWorK, or send me an email)

Lesson 12: Polynomial and Rational Inequalities

Hi everyone! Read through the material below, watch the videos, and send me your questions.

Lesson 12: Polynomial and Rational Inequalities

Topic: This lesson covers Chapter 12 in the book, Polynomial and Rational Inequalities.

WeBWorK: There are two WeBWorK assignments on today’s material: Polynomials - Inequalities, and Rational Functions - Inequalities.

Introduction to polynomial inequalities

Definition. A polynomial inequality is an inequality (which means it uses one of these: <,\leq,>,\geq instead of an equals sign) with a polynomial on each side

Example 1: x^2-3x-4\geq 0

We’re interested in solving these inequalities, which means answering the question: “For which real numbers x is the inequality true?”

Now let’s look at the same example, and see how to solve it without looking at the graph:

Example 2: Solve x^{4}-x^{2}>5\left(x^{3}-x\right)

Example 2, concluded:

Rational inequalities

What happens if we allow rational functions instead of just polynomials?

Example 3: Solve \frac{x^{2}-5 x+6}{x^{2}-5 x} \geq 0

Good job! You are now ready to practice on your own. Take a look at the WeBWorK assignment, and don’t forget to use the “Ask for Help” button if you get stuck.

Here are more video resources if you’d like to see additional examples.

ASSIGNMENT: Watch videos, try webwork.