A sequence is a function whose inputs are the positive integers. An arithmetic sequence adds a common difference d each term, 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). Each can also be written recursively from the previous term.
Key facts:
- Standards F-IF.A.3, F-BF.A.2 and F-LE.A.2: sequences as functions, explicit and recursive formulas, and modeling with arithmetic and geometric sequences.
- Arithmetic: \(a_{n}=a_{1}+(n-1)d\); recursive \(a_{n}=a_{n-1}+d\).
- Geometric: \(a_{n}=a_{1}\cdot r^{n-1}\); recursive \(a_{n}=r\cdot a_{n-1}\).
- Arithmetic sequences grow linearly; geometric sequences grow or decay exponentially.
- To identify the type, test consecutive differences first, then consecutive ratios.
What is a sequence?
A sequence is a function whose inputs are the positive integers \(1,2,3,\) and so on. The input \(n\) is the position and the output \(a_{n}\) is the term. The first term is \(a_{1}\), the second is \(a_{2}\), and \(a_{n}\) is the \(n\)th term. Because the inputs are whole numbers, the graph is a set of separate points, not a connected line.
What is an arithmetic sequence?
An arithmetic sequence adds the same number each time. That number is the common difference \(d\).
The sequence \(3,7,11,15,19\) has \(d=4\). Its recursive formula gives each term from the one before: \(a_{1}=3\) and \(a_{n}=a_{n-1}+4\) for \(n\ge 2\). Its explicit formula gives any term directly: \(a_{n}=a_{1}+(n-1)d=3+4(n-1)\), so \(a_{20}=3+4(19)=79\). The graph is a line, so an arithmetic sequence is a linear function.
What is a geometric sequence?
A geometric sequence multiplies by the same number each time. That number is the common ratio \(r\).
The sequence \(2,6,18,54\) has \(r=3\). Recursive: \(a_{1}=2\) and \(a_{n}=3\cdot a_{n-1}\). Explicit: \(a_{n}=a_{1}\cdot r^{n-1}=2\cdot 3^{n-1}\), so \(a_{6}=2\cdot 3^{5}=486\). A geometric sequence is an exponential function of \(n\).
How do you tell the two types apart and write a formula?
- Subtract consecutive terms. If the differences are all equal, it is arithmetic.
- Divide consecutive terms. If the ratios are all equal, it is geometric.
- Write the formula with \(a_{1}\) and \(d\) or \(r\).
- Check one term you already know.
Worked example. For \(5,10,20,40\) the differences are \(5,10,20\) (not equal) but every ratio is 2, so it is geometric with \(a_{n}=5\cdot 2^{n-1}\). Check: \(a_{4}=5\cdot 8=40\).
How do sequences model real situations?
A savings jar that gets \(\$6\) added each week is arithmetic. A video whose views double daily is geometric. Linear growth adds the same amount each step, while exponential growth multiplies, so a geometric sequence with \(r>1\) eventually passes any arithmetic one.
Arithmetic and Geometric Sequences: frequently asked questions
What is the difference between an arithmetic and a geometric sequence?
An arithmetic sequence adds the same number each time, such as \(4,7,10,13\). A geometric sequence multiplies by the same number each time, such as \(4,8,16,32\).
What is the difference between a recursive and an explicit formula?
A recursive formula needs the previous term and a starting value, such as \(a_{n}=a_{n-1}+4\) with \(a_{1}=3\). An explicit formula gives the \(n\)th term directly, such as \(a_{n}=3+4(n-1)\).
How do you find the 25th term of an arithmetic sequence?
Use \(a_{n}=a_{1}+(n-1)d\). For \(a_{1}=7\) and \(d=3\), \(a_{25}=7+24(3)=79\). You do not need to list all 25 terms.
How do you find the common ratio?
Divide any term by the term before it. For \(6,18,54\), \(18: 6=3\) and \(54: 18=3\), so \(r=3\). A ratio can be a fraction or negative.
Why is a sequence a function?
Each position \(n\) has exactly one term \(a_{n}\). The domain is the positive integers, so the graph is a set of separate points.
See every chapter on the Algebra 1 math page.