Exponential Modeling and Logarithms

Lesson 5

Math

Unit 5

11th Grade

Lesson 5 of 16

Objective


Describe the derivation of the formula for the sum of a finite geometric series and use it to solve problems.

Common Core Standards


Core Standards

  • 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.

Criteria for Success


  1. Identify the difference between a sequence and a series.
  2. Describe how the formula for the sum of a geometric series can be derived.
  3. Rewrite sums using sigma notation.
  4. Evaluate the sum of a geometric series.

Tips for Teachers


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Anchor Problems


Problem 1

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:

$${S=a+ar+ar^2+...+ar^{n-1}}$$

$${S=a(1+r+r^2+...+r^{n-1})}$$

$${S(1-r)=a(1-r^n)}$$

$${S=a\left ( {1-r^n\over{1-r}} \right )}$$

Guiding Questions

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Problem 2

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.

Guiding Questions

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Problem 3

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:

$${S(n)=a\left ( \frac{1-r^n}{1-r} \right )}$$

Guiding Questions

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Target Task


Problem 1

 Find the sum of the series below. 

$${\sum_{n=1}^{4}(-1)\left ( 1\over2 \right )^{n-1}}$$

Problem 2

Write the following series using summation notation, then find the sum of the series.

$${{1\over3}+{1\over9}+{1\over27}+{1\over81}}$$

Problem 3

Find the value of each of the sums in #1 and #2 using the formula for the sum of a geometric series.

Additional Practice


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.

  • Include problems finding sums from sigma notation, series as lists, or a certain number of terms from a function. Include both growth and decay sequences.
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Lesson 4

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Lesson 6

Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Modeling with and Interpreting Exponential Functions

Topic B: Definition and Meaning of Logarithms

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