Precalculus Sequences and Series Lesson (Free PDF)
This precalculus sequences series lesson for Precalculus covers sequences and series (Common Core F-IF.A.3, F-BF.A.2, A-SSE.B.4), with a free PDF to print.
A sequence is an ordered list of numbers. An arithmetic sequence adds a common difference d, so a_n = a_1 + (n – 1)d. A geometric sequence multiplies by a common ratio r, so a_n = a_1 r^(n-1). An infinite geometric series with |r| < 1 has the sum a_1/(1 - r).
Precalculus · Common Core State Standards: F-IF.A.3, F-BF.A.2, A-SSE.B.4
Key facts in this precalculus sequences series lesson
- Arithmetic: a_n = a_1 + (n – 1)d; sum S_n = (n/2)(a_1 + a_n).
- Geometric: a_n = a_1 r^(n-1); sum S_n = a_1(1 – r^n)/(1 – r).
- Sigma notation adds the terms from the lower index to the upper index.
- An infinite geometric series converges only if |r| < 1, and then S = a_1/(1 - r).
- A recursive formula gives each term from the one before it.
What is a sequence?
A sequence is an ordered list of numbers \(a_{1},a_{2},a_{3},\text{ … }\). An explicit formula gives \(a_{n}\) directly from \(n\), and a recursive formula gives each term from the previous one. A sequence is a function whose domain is the positive integers, so its graph is a set of separate points.
| Type | Pattern | Explicit formula | Recursive formula |
|---|---|---|---|
| Arithmetic | add the common difference \(d\) | \(a_{n}=a_{1}+(n-1)d\) | \(a_{n}=a_{n-1}+d\) |
| Geometric | multiply by the common ratio \(r\) | \(a_{n}=a_{1}r^{n-1}\) | \(a_{n}=r\,a_{n-1}\) |
To decide which type you have, test the pattern. Subtract each term from the next: if every difference is the same, the sequence is arithmetic. Divide each term by the one before it: if every ratio is the same, it is geometric. For \(5,10,20,40\) the differences are \(5,10,20\), which are not equal, but every ratio is 2. Many real situations follow these patterns: a salary that rises by a fixed raise each year is arithmetic, while a savings account earning a fixed percent or a ball that rebounds to a fixed fraction of its height is geometric. A recursive formula such as \(a_{n}=a_{n-1}+4\) with \(a_{1}=5\) describes the same arithmetic sequence as the explicit formula \(a_{n}=4n+1\).
What is a series and how do you sum it?
A series is the sum of the terms of a sequence. Sigma notation \(\sum_{k=1}^{n}a_{k}\) means \(a_{1}+a_{2}+\text{ … }+a_{n}\), where \(k\) runs from the bottom number to the top number. For example \(\sum_{k=1}^{5}(2k+1)=3+5+7+9+11=35\).
The arithmetic formula pairs the first and last terms: each pair sums to \(a_{1}+a_{n}\), and there are \(\frac n2\) pairs. For \(7+11+15+\text{ … }\) with 20 terms, \(a_{20}=83\) and \(S_{20}=10(7+83)=900\).
When does an infinite series have a sum?
An infinite geometric series converges only when \(|r|< 1\), and then \(S=\frac{a_{1}}{1-r}\). For \(8+4+2+1+\text{ ... }\), \(a_{1}=8\) and \(r=\frac12\), so \(S=\frac{8}{1/2}=16\). If \(|r|\ge1\), the partial sums do not settle down and the series diverges.
Infinite series explain repeating decimals. Since \(0.\overline{36}=0.36+0.0036+0.000036+\text{ … }\) is geometric with \(a_{1}=0.36\) and \(r=0.01\), its sum is \(\frac{0.36}{0.99}=\frac{4}{11}\).
Precalculus sequences series lesson: questions and answers
How do you tell if a sequence is arithmetic or geometric?
Subtract consecutive terms: if the differences are constant, it is arithmetic. Divide consecutive terms: if the ratios are constant, it is geometric.
What does sigma notation mean?
The symbol sigma means sum. The sum from k = 1 to 5 of (2k + 1) adds the values of 2k + 1 for k = 1, 2, 3, 4, 5.
When does an infinite geometric series have a sum?
When the absolute value of the ratio is less than 1. Then the terms shrink fast enough, and the sum is the first term divided by 1 minus r.
What is the difference between a sequence and a series?
A sequence is a list of numbers, such as 2, 4, 8, 16. A series is the sum of those terms, such as 2 + 4 + 8 + 16 = 30.
What is the difference between explicit and recursive formulas?
An explicit formula finds any term directly from n. A recursive formula needs the previous term, so you must start from the first term and work forward.
How to use this precalculus 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 precalculus sequences series lesson as a PDF with the button below and keep it with your notes.
Download the precalculus sequences series lesson as a PDF with the button below and keep it with your notes.
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