Curriculum / Math / 11th Grade / Unit 5: Exponential Modeling and Logarithms / Lesson 5
Math
Unit 5
11th Grade
Lesson 5 of 16
Jump To
Lesson Notes
There was an error generating your document. Please refresh the page and try again.
Generating your document. This may take a few seconds.
Are you sure you want to delete this note? This action cannot be undone.
Describe the derivation of the formula for the sum of a finite geometric series and use it to solve problems.
The core standards covered in this lesson
A.SSE.B.4 — Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
The essential concepts students need to demonstrate or understand to achieve the lesson objective
Suggestions for teachers to help them teach this lesson
Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.
Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
25-30 minutes
Find the following products:
$${(1-r)(1+r)}$$
$${(1-r)(1+r+r^2)}$$
$${(1-r)(1+r+r^2+r^3)}$$
What happens in each case? What happens for the general product $${(1-r)(1+r+r^2+...+r^n)}$$?
Explain what is happening at each step:
$${S=a+ar+ar^2+...+ar^{n-1}}$$
$${S=a(1+r+r^2+...+r^{n-1})}$$
$${S(1-r)=a(1-r^n)}$$
$${S=a\left ( {1-r^n\over{1-r}} \right )}$$
This "summation" or "sigma" notation:
$${\sum_{n=1}^{5}2\left ( 1\over2 \right )^{n-1}}$$
Describes this:
$${2+1+{1\over2}+{1\over4}+{1\over8}}$$
Explain what all the components of the "sigma notation" are.
Find the value of:
$${\sum_{n=1}^{5}2\left ( {1\over2} \right )^{n-1}=}$$
Then use the sum of a finite geometric series formula to get the same value:
$${S(n)=a\left ( \frac{1-r^n}{1-r} \right )}$$
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
5-10 minutes
Find the sum of the series below.
$${\sum_{n=1}^{4}(-1)\left ( 1\over2 \right )^{n-1}}$$
Write the following series using summation notation, then find the sum of the series.
$${{1\over3}+{1\over9}+{1\over27}+{1\over81}}$$
Find the value of each of the sums in #1 and #2 using the formula for the sum of a geometric series.
The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.
Next
Find the sum of an infinite geometric series.
Topic A: Modeling with and Interpreting Exponential Functions
Identify, model, and analyze geometric sequences.
Standards
F.IF.A.3F.IF.B.5F.LE.A.2
Analyze and construct exponential functions that model contexts.
F.IF.B.4F.IF.C.8.BF.LE.A.2
Write and change the form of exponential functions that model compounding interest.
F.BF.A.1.AF.LE.B.5
Define and use $$e$$ in continuous compounding situations.
A.SSE.B.3.CF.BF.A.1.A
A.SSE.B.4
Create a free account to access thousands of lesson plans.
Already have an account? Sign In
Topic B: Definition and Meaning of Logarithms
Describe and evaluate simple numeric logarithms (Part I).
F.LE.A.4
Describe and evaluate simple numeric logarithms (Part II).
Describe logarithms as the inverse of exponential functions and graph logarithmic functions.
F.BF.B.3F.BF.B.4.BF.BF.B.4.CF.BF.B.5F.IF.C.7.E
Evaluate common and natural logs using tables, graphs, and calculators.
F.BF.B.4.CF.LE.A.4
Understand and apply the change of base property to evaluate logarithms.
Develop and use the product and quotient properties of logarithms to write equivalent expressions.
Develop and use the power property of logarithms to write equivalent expressions.
F.BF.B.4.BF.LE.A.4
Solve equations with logarithms.
Use logarithms to solve exponential modeling problems (Part I).
A.SSE.A.1.BF.LE.A.4
Use logarithms to solve exponential modeling problems (Part II).
See all of the features of Fishtank in action and begin the conversation about adoption.
Learn more about Fishtank Learning School Adoption.
Yes
No
We've got you covered with rigorous, relevant, and adaptable math lesson plans for free