Algebra 2 Arithmetic and Geometric Sequences and Series Lesson (Free PDF)
This algebra 2 sequences series lesson for Algebra 2 covers arithmetic and geometric sequences and series (Common Core F-BF.A.2, A-SSE.B.4, F-IF.A.3), with a free PDF to print.
A sequence is a function on the positive integers. An arithmetic sequence adds a common difference d and has a_n = a_1 + (n – 1)d, while a geometric sequence multiplies by a common ratio r and has a_n = a_1·r^(n-1). The sum of the first n terms is called a series, and each type has a short sum formula.
Algebra 2 · Common Core State Standards: F-BF.A.2, A-SSE.B.4, F-IF.A.3
Key facts in this algebra 2 sequences series lesson
- A sequence is a function whose domain is the positive integers.
- Arithmetic: a_n = a_1 + (n – 1)d; geometric: a_n = a_1·r^(n-1).
- Recursive formulas build each term from the previous one.
- Arithmetic sum: S_n = n(a_1 + a_n)/2.
- Geometric sum: S_n = a_1(1 – r^n)/(1 – r), r ≠ 1.
What is a sequence?
A sequence is a function whose domain is the positive integers: \(a_{1}, a_{2}, a_{3}, …\) written here as a list of terms \(a_{n}\). A sequence can be given by an explicit formula such as \(a_{n} = 3n – 2\), or by a recursive formula that tells how to get each term from the previous ones, such as \(a_{1} = 5\) and \(a_{n} = a_{n-1} + 4\).
Arithmetic sequences are linear and geometric sequences are exponential. For \(7, 11, 15, 19, …\), \(d = 4\), so \(a_{n} = 7 + (n – 1)4 = 4n + 3\). For \(80, 40, 20, 10, …\), \(r = \frac{1}{2}\), so \(a_{n} = 80\left(\frac{1}{2}\right)^{n-1}\).
How do you add the terms of a series?
A series is the sum of the terms of a sequence. Let \(S_{n}\) be the sum of the first \(n\) terms. For an arithmetic series, pair the first and last terms: \(S_{n} = \frac{n(a_{1} + a_{n})}{2}\). For a geometric series with \(r \ne 1\): \(S_{n} = a_{1}\cdot\frac{1 – r^{n}}{1 – r}\).
- Decide whether the sequence is arithmetic or geometric.
- Find \(a_{1}\), the number of terms \(n\), and \(d\) or \(r\).
- Choose the right sum formula and substitute.
Example: \(3 + 6 + 12 + …\) with 8 terms has \(a_{1} = 3\), \(r = 2\), so \(S_{8} = 3\cdot\frac{1 – 2^{8}}{1 – 2} = 3 \cdot 255 = 765\). A geometric series is the model for savings plans: if you deposit $200 every year at 5%, the balance after 10 years is a geometric series.
Algebra 2 sequences series lesson: questions and answers
What is the difference between an arithmetic and a geometric sequence?
An arithmetic sequence adds the same number each step, like 3, 7, 11. A geometric sequence multiplies by the same number each step, like 3, 6, 12.
What is the explicit formula for an arithmetic sequence?
a_n = a_1 + (n – 1)d, where a_1 is the first term and d is the common difference.
What is the sum formula for a geometric series?
S_n = a_1 (1 – r^n) / (1 – r) for r not equal to 1, where a_1 is the first term, r is the common ratio, and n is the number of terms.
What is a recursive formula?
A recursive formula gives the first term and a rule that builds each term from the previous one, for example a_1 = 2 and a_n = 3 a_(n-1).
Why is a sequence a function?
Each position number n has exactly one term a_n. The domain is the positive integers, so a sequence is a function whose graph is a set of separate points.
How to use this algebra 2 sequences series lesson
Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.
Download the algebra 2 sequences series lesson as a PDF with the button below and keep it with your notes.
Download the algebra 2 sequences series lesson as a PDF with the button below and keep it with your notes.
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