Week 8, Session 1: Arithmetic and Geometric Sequences

Session Focus

Today you will represent arithmetic and geometric sequences explicitly and recursively. This session is written for independent study, but it should feel like a teacher is sitting beside you and asking the right questions at the right time. Read slowly, keep paper beside you, and do each example before you look at the explanation.

Key Language

sequence, arithmetic, geometric, common difference, common ratio

Teacher Explanation

A sequence is a function whose inputs are usually positive integers. Arithmetic sequences add the same amount; geometric sequences multiply by the same amount. This is a high-difficulty course, so the goal is not only to get an answer. The goal is to understand why the method works, what restrictions are present, and how the same idea appears in symbolic, graphical, numerical, and verbal forms.

When you study this session, use a three-pass routine. On the first pass, read the explanation and copy the main rule in your own words. On the second pass, redo the worked examples from a blank page. On the third pass, complete the independent practice without checking notes until you have a full attempt.

Visual Study Cue

Use the image above as a quick map of the session. Before solving, name the main object in the visual, identify what is changing, and connect the picture to the rule below. This habit helps you move between graphical, symbolic, numerical, and verbal reasoning.

Core Rule

Arithmetic explicit form is \(a_n = a_1 + (n-1)d\); geometric explicit form is \(a_n = a_1 r^{n-1}\).

Worked Example 1: Writing explicit formulas

For the arithmetic sequence \(7, 11, 15, 19, ...\), the first term is 7 and the common difference is 4. The explicit formula is \(a_n = 7 + (n - 1)4\), which simplifies to \(a_n = 4n + 3\). The 25th term is \(4(25)+3 = 103\).

Pause after the example and ask yourself what made the solution move forward. Was it a definition, a theorem, a graph feature, an algebraic manipulation, or a domain restriction? Naming the reason matters because it helps you transfer the method to harder problems.

Worked Example 2: A second angle on the same idea

For the geometric sequence \(3, 12, 48, 192, ...\), the first term is 3 and the common ratio is 4. The explicit formula is \(a_n = 3 \cdot 4^{n - 1}\). The fifth term is \(3 \cdot 4^4 = 768\).

Notice how the second example uses the same core idea but changes the surface details. That is deliberate. Advanced algebra and precalculus become difficult when a familiar idea is hidden inside a new representation. Your task is to recognize the structure underneath the wording.

Independent Practice

• Write explicit and recursive formulas for \(10, 7, 4, 1, ...\).
• Find the 8th term of \(2, 6, 18, ...\).
• Explain why sequence input usually begins at n=1.

Error-Log Reflection

After you finish, write one sentence beginning with "The step I must watch most carefully is..." Then write one sentence beginning with "The clue that tells me to use this method is..." These two sentences convert practice into self-learning because they make your decision process visible.

References

• Abramson, J. (2021). Algebra and trigonometry 2e. OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e
• College Board. (2023). AP Precalculus course and exam description. https://apcentral.collegeboard.org/media/pdf/ap-precalculus-course-and-exam-description.pdf
• National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. https://corestandards.org/wp-content/uploads/2023/09/Math_Standards1.pdf