The Central Idea
Consider the sequence
\[ 3,\ 5,\ 7,\ 9,\ 11,\ldots \]If indexing begins at \(1\), then
\[ a_1=3,\qquad a_2=5,\qquad a_3=7,\qquad a_4=9,\qquad a_5=11. \]We can organize this information in a table:
| Index \(n\) | Term \(a_n\) |
|---|---|
| \(1\) | \(3\) |
| \(2\) | \(5\) |
| \(3\) | \(7\) |
| \(4\) | \(9\) |
| \(5\) | \(11\) |
The same information can be written as ordered pairs:
\[ \boxed{ (1,3),\ (2,5),\ (3,7),\ (4,9),\ (5,11) }. \]Plotting those ordered pairs gives a discrete graph of the sequence.
A Sequence as Input and Output
A numerical sequence can be viewed as a function whose inputs are discrete indices.
The input is
\[ n, \]and the corresponding output is
\[ a_n. \]Thus the basic relationship is
\[ \boxed{n\longmapsto a_n}. \]A sequence table is simply a function table adapted to discrete indexing.
The Two Columns of a Sequence Table
A standard sequence table contains:
- the index in one column;
- the corresponding term value in another.
For
\[ a_n=n^2, \]we obtain:
| \(n\) | \(a_n\) |
|---|---|
| \(1\) | \(1\) |
| \(2\) | \(4\) |
| \(3\) | \(9\) |
| \(4\) | \(16\) |
| \(5\) | \(25\) |
From a Formula to a Table
Suppose
\[ a_n=4n-1,\qquad n\geq1. \]Evaluate the formula at successive indices:
\[ a_1=4(1)-1=3, \] \[ a_2=4(2)-1=7, \] \[ a_3=4(3)-1=11, \] \[ a_4=4(4)-1=15. \]The table is:
| \(n\) | \(a_n\) |
|---|---|
| \(1\) | \(3\) |
| \(2\) | \(7\) |
| \(3\) | \(11\) |
| \(4\) | \(15\) |
From a Table to a Listed Sequence
Suppose a table gives:
| \(n\) | \(a_n\) |
|---|---|
| \(0\) | \(5\) |
| \(1\) | \(8\) |
| \(2\) | \(11\) |
| \(3\) | \(14\) |
Because indexing begins at \(0\), the sequence is
\[ \boxed{5,\ 8,\ 11,\ 14,\ldots}. \]We also read
\[ a_0=5,\qquad a_1=8,\qquad a_2=11. \]From a Table to Ordered Pairs
Every row of a sequence table corresponds to an ordered pair
\[ (n,a_n). \]For example,
| \(n\) | \(a_n\) |
|---|---|
| \(1\) | \(4\) |
| \(2\) | \(9\) |
| \(3\) | \(14\) |
| \(4\) | \(19\) |
becomes
\[ \boxed{ (1,4),\ (2,9),\ (3,14),\ (4,19) }. \]What Is a Discrete Graph?
A discrete graph consists of separated plotted points corresponding to the allowed inputs.
For a sequence, the points have the form
\[ \boxed{(n,a_n)}. \]The horizontal coordinate represents the index, and the vertical coordinate represents the term value.
Why Sequence Graphs Are Discrete
Suppose a sequence is defined for
\[ n=1,2,3,\ldots \]Then the allowed horizontal coordinates are the positive integers.
Values such as
\[ n=1.5 \]are not sequence indices unless they have been explicitly included in the domain.
Therefore, the graph should normally display separate points rather than a continuous curve.
Do Not Automatically Connect the Points
Consider
\[ a_n=n^2,\qquad n=1,2,3,\ldots \]The plotted sequence points are
\[ (1,1),\ (2,4),\ (3,9),\ (4,16),\ldots \]Joining these points with a continuous parabola would suggest outputs for inputs such as
\[ 2.5, \]even though \(2.5\) is not an allowed sequence index.
The parabola may be a useful continuous extension, but it is not identical to the sequence.
Discrete Domain, Familiar Formula
Compare
\[ f(x)=2x+1,\qquad x\in\mathbb{R} \]with
\[ a_n=2n+1,\qquad n=1,2,3,\ldots \]The formulas have the same algebraic structure.
But their domains differ:
- \(f\) may accept every real input;
- \(a_n\) accepts only the specified integer indices.
Domain is part of the mathematical object.
Arithmetic Sequences in Tables
Consider
\[ a_n=3n+2. \]| \(n\) | \(a_n\) | First Difference |
|---|---|---|
| \(1\) | \(5\) | \(3\) |
| \(2\) | \(8\) | \(3\) |
| \(3\) | \(11\) | \(3\) |
| \(4\) | \(14\) | \(3\) |
| \(5\) | \(17\) |
Equal index steps produce equal changes in the term values.
Arithmetic Sequences on a Graph
For
\[ a_n=3n+2, \]the sequence points are
\[ (1,5),\ (2,8),\ (3,11),\ (4,14),\ldots \]These points lie on a straight line.
The common difference \(3\) appears graphically as a constant vertical change for each one-unit horizontal step.
Thus arithmetic sequences are discrete linear relationships.
Common Difference as Discrete Slope
Between consecutive sequence points,
\[ (n,a_n) \]and
\[ (n+1,a_{n+1}), \]the slope is
\[ \frac{a_{n+1}-a_n}{(n+1)-n}. \]Since the denominator is \(1\),
\[ \boxed{ \text{slope}=a_{n+1}-a_n=d }. \]The common difference is therefore the slope between successive plotted points.
Geometric Sequences in Tables
Consider
\[ a_n=2\cdot3^{n-1}. \]| \(n\) | \(a_n\) | Ratio to Previous Term |
|---|---|---|
| \(1\) | \(2\) | |
| \(2\) | \(6\) | \(3\) |
| \(3\) | \(18\) | \(3\) |
| \(4\) | \(54\) | \(3\) |
| \(5\) | \(162\) | \(3\) |
Equal index steps produce equal multiplicative changes.
Geometric Sequences on a Graph
The sequence
\[ a_n=2\cdot3^{n-1} \]produces
\[ (1,2),\ (2,6),\ (3,18),\ (4,54),\ldots \]These points lie on an exponential curve.
Unlike an arithmetic sequence, the vertical changes are not constant.
Instead, the term values are repeatedly multiplied by the same factor.
Quadratic Sequences on a Graph
Consider
\[ a_n=n^2. \]The points
\[ (1,1),\ (2,4),\ (3,9),\ (4,16),\ldots \]lie on a parabola.
The first differences change, but the second differences are constant.
The graph bends rather than following a straight-line pattern.
Visual Patterns and Algebraic Patterns
Different sequence families produce characteristic visual behavior.
| Sequence Type | Algebraic Feature | Graphical Pattern |
|---|---|---|
| Constant | \(a_n=c\) | Horizontal row of points |
| Arithmetic | Constant first difference | Points on a line |
| Quadratic | Constant second difference | Points on a parabola |
| Geometric | Constant ratio | Points on an exponential curve |
Recursive Sequences in Tables
A table is especially useful for recursive sequences because each completed row supplies information needed later.
Suppose
\[ a_1=2,\qquad a_n=2a_{n-1}+1. \]Then:
| \(n\) | Calculation | \(a_n\) |
|---|---|---|
| \(1\) | Initial value | \(2\) |
| \(2\) | \(2(2)+1\) | \(5\) |
| \(3\) | \(2(5)+1\) | \(11\) |
| \(4\) | \(2(11)+1\) | \(23\) |
| \(5\) | \(2(23)+1\) | \(47\) |
Fibonacci-Type Sequences in Tables
For
\[ F_0=0,\qquad F_1=1,\qquad F_n=F_{n-1}+F_{n-2}, \]a table makes the recursive dependency easy to follow:
| \(n\) | \(F_n\) |
|---|---|
| \(0\) | \(0\) |
| \(1\) | \(1\) |
| \(2\) | \(1\) |
| \(3\) | \(2\) |
| \(4\) | \(3\) |
| \(5\) | \(5\) |
| \(6\) | \(8\) |
Tables Can Reveal Missing Values
Suppose an arithmetic sequence is represented by:
| \(n\) | \(a_n\) |
|---|---|
| \(1\) | \(7\) |
| \(2\) | \(12\) |
| \(3\) | \(\square\) |
| \(4\) | \(22\) |
The common difference is
\[ 12-7=5. \]Therefore,
\[ a_3=12+5=\boxed{17}. \]Tables Can Reveal a Rule
Suppose
| \(n\) | \(a_n\) |
|---|---|
| \(1\) | \(6\) |
| \(2\) | \(10\) |
| \(3\) | \(14\) |
| \(4\) | \(18\) |
The outputs increase by \(4\).
Thus,
\[ d=4. \]Since \(a_1=6\),
\[ a_n=6+4(n-1) \] \[ =\boxed{4n+2}. \]Graph to Table
If a discrete graph contains the points
\[ (0,4),\ (1,7),\ (2,10),\ (3,13), \]then its sequence table is:
| \(n\) | \(a_n\) |
|---|---|
| \(0\) | \(4\) |
| \(1\) | \(7\) |
| \(2\) | \(10\) |
| \(3\) | \(13\) |
The graph and the table encode exactly the same ordered pairs.
Graph to Sequence Notation
From the points
\[ (1,5),\ (2,10),\ (3,15),\ (4,20), \]we read
\[ a_1=5,\qquad a_2=10,\qquad a_3=15,\qquad a_4=20. \]The common difference is \(5\), so an explicit rule is
\[ \boxed{a_n=5n}. \]Starting Index and Horizontal Position
The starting index determines where the first plotted point appears horizontally.
If
\[ a_0=5, \]then the first point is
\[ (0,5). \]If instead
\[ b_1=5, \]then the first point is
\[ (1,5). \]The displayed value \(5\) is the same, but its indexed position differs.
Finite Sequences on a Graph
A finite sequence produces only finitely many plotted points.
For example,
\[ a_n=2n,\qquad 1\leq n\leq5 \]gives
\[ (1,2),\ (2,4),\ (3,6),\ (4,8),\ (5,10). \]There are no sequence points beyond \(n=5\).
Infinite Sequences on a Graph
An infinite sequence has points at arbitrarily large allowed indices.
For
\[ a_n=\frac1n,\qquad n\geq1, \]the graph contains
\[ \left(1,1\right), \left(2,\frac12\right), \left(3,\frac13\right), \left(4,\frac14\right), \ldots \]The points continue indefinitely to the right.
Graphs Can Suggest Long-Run Behavior
Consider
\[ a_n=\frac1n. \]As the index grows,
\[ a_n \]gets closer to \(0\).
A discrete graph makes this tendency visible because the points descend toward the horizontal axis.
Later, this behavior will be described formally using convergence.
Oscillating Sequences on a Graph
Consider
\[ a_n=(-1)^n. \]Beginning at \(n=1\), the values are
\[ -1,\ 1,\ -1,\ 1,\ldots \]The graph alternates between the horizontal levels
\[ y=-1 \]and
\[ y=1. \]Discrete graphs are especially useful for making oscillation visible.
Negative Term Values
Sequence points may lie below the horizontal axis.
For
\[ a_n=5-2n, \]the first terms are
\[ 3,\ 1,\ -1,\ -3,\ldots \]corresponding to
\[ (1,3),\ (2,1),\ (3,-1),\ (4,-3),\ldots \]The graph crosses from positive to negative term values while remaining discrete.
Repeated Values on a Graph
Different indices may have the same output.
For
\[ 2,\ 5,\ 2,\ 5,\ 2,\ldots, \]several plotted points share the same vertical coordinate.
For example,
\[ (1,2),\ (3,2),\ (5,2) \]all lie on the line \(y=2\).
This does not violate the definition of a sequence because different indices may map to the same value.
A Sequence Still Passes the Vertical-Line Test
Because each allowed index has only one assigned term value, a sequence viewed as a graph is a function.
At any allowed horizontal coordinate \(n\), there is only one point
\[ (n,a_n). \]Repeated outputs are permitted because several different inputs may share the same output.
Tables and Difference Patterns
Tables can be expanded to include finite differences.
For
\[ a_n=n^2, \]we may write:
| \(n\) | \(a_n\) | First Difference | Second Difference |
|---|---|---|---|
| \(1\) | \(1\) | \(3\) | \(2\) |
| \(2\) | \(4\) | \(5\) | \(2\) |
| \(3\) | \(9\) | \(7\) | \(2\) |
| \(4\) | \(16\) | \(9\) | |
| \(5\) | \(25\) |
The table reveals quadratic structure numerically.
Tables and Ratios
For geometric sequences, a ratio column may be more informative than a difference column.
Consider
\[ 5,\ 10,\ 20,\ 40,\ldots \]| \(n\) | \(a_n\) | \(a_n/a_{n-1}\) |
|---|---|---|
| \(1\) | \(5\) | |
| \(2\) | \(10\) | \(2\) |
| \(3\) | \(20\) | \(2\) |
| \(4\) | \(40\) | \(2\) |
The constant ratio reveals the geometric rule.
Representations Should Agree
If a sequence is described by a formula, table, graph, and recurrence, all representations must encode the same index-value pairs.
For example,
\[ a_n=2n+1 \]must agree with the table entry
\[ n=3,\qquad a_3=7 \]and with the plotted point
\[ (3,7). \]Translating among representations is also a powerful checking method.
Choosing a Useful Representation
Different representations emphasize different information.
- A list shows term order quickly.
- A formula may give direct access to distant terms.
- A recurrence shows how terms generate later terms.
- A table makes index-value correspondence explicit.
- A graph reveals visual trends, growth, decay, and oscillation.
Good mathematical problem solving often involves moving deliberately among several representations.
Common Error: Reversing the Coordinates
A sequence point is normally written
\[ (n,a_n). \]The index belongs on the horizontal axis, and the term value belongs on the vertical axis.
If
\[ a_3=10, \]the point is
\[ \boxed{(3,10)}, \]not \((10,3)\).
Common Error: Connecting All Sequence Points
Connecting points can incorrectly imply that intermediate horizontal values belong to the domain.
If the sequence is defined only at integer indices, plot those integer-index points separately.
A continuous curve may be shown separately when discussing a continuous extension, but the distinction should remain clear.
Common Error: Ignoring the Starting Index
Suppose a table begins
| \(n\) | \(a_n\) |
|---|---|
| \(0\) | \(2\) |
| \(1\) | \(5\) |
| \(2\) | \(8\) |
The first listed term is
\[ a_0=2, \]not \(a_1=2\).
The index column controls the notation.
Common Error: Reading Equal Differences from Unequal Index Steps
Suppose a table uses indices
\[ 1,\ 2,\ 4,\ 7. \]The horizontal steps are
\[ 1,\ 2,\ 3. \]Simply comparing vertical differences without accounting for these unequal index changes can give a misleading impression of rate.
Standard arithmetic and finite-difference tests assume equal input spacing.
Common Error: Treating a Graph as More Precise Than Its Data
A graph can make patterns visible, but exact values should be read from labels, coordinates, tables, or formulas when available.
Visual estimation is useful, but it should not replace exact calculation when exact sequence values are required.
A Reliable Procedure for Building a Sequence Table
- Identify the starting index.
- List the required index values in order.
- Evaluate the explicit rule or recurrence at each index.
- Record exactly one term value for each index.
- Check that the values follow the stated sequence rule.
- Add difference or ratio columns when they help reveal structure.
A Reliable Procedure for Graphing a Sequence
- Construct or read the sequence table.
- Convert each row into an ordered pair \((n,a_n)\).
- Place the index on the horizontal axis.
- Place the term value on the vertical axis.
- Plot only the allowed indices.
- Do not connect the points unless a separate continuous interpretation is intended.
- Inspect the graph for growth, decay, oscillation, linearity, or other patterns.
Example: Full Table-and-Graph Workflow
Let
\[ a_n=5-2n,\qquad n=0,1,2,3,4. \]Evaluate:
\[ a_0=5, \] \[ a_1=3, \] \[ a_2=1, \] \[ a_3=-1, \] \[ a_4=-3. \]The table is:
| \(n\) | \(a_n\) |
|---|---|
| \(0\) | \(5\) |
| \(1\) | \(3\) |
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(4\) | \(-3\) |
The plotted points are
\[ \boxed{ (0,5),\ (1,3),\ (2,1),\ (3,-1),\ (4,-3) }. \]They lie on a descending line, but the sequence consists only of these five discrete points.
Example: Recognizing a Geometric Pattern from a Table
| \(n\) | \(a_n\) |
|---|---|
| \(1\) | \(3\) |
| \(2\) | \(6\) |
| \(3\) | \(12\) |
| \(4\) | \(24\) |
Consecutive ratios are
\[ \frac63=2, \qquad \frac{12}{6}=2, \qquad \frac{24}{12}=2. \]Therefore,
\[ r=2. \]Since \(a_1=3\),
\[ \boxed{a_n=3\cdot2^{n-1}}. \]Practice 1: Formula to Table
Build a table for
\[ a_n=3n+1 \]for
\[ n=1,2,3,4. \]Practice 2: Table to Ordered Pairs
Convert the following table into ordered pairs:
| \(n\) | \(a_n\) |
|---|---|
| \(0\) | \(6\) |
| \(1\) | \(9\) |
| \(2\) | \(12\) |
| \(3\) | \(15\) |
Practice 3: Ordered Pairs to Sequence
The graph of a sequence contains
\[ (1,4),\ (2,8),\ (3,12),\ (4,16). \]Write the listed sequence and an explicit formula.
Practice 4: Identify the Sequence Type
A table gives
\[ 5,\ 10,\ 20,\ 40,\ldots \]as the output values at consecutive integer indices.
Is the sequence arithmetic, geometric, or neither?
Practice 5: Starting at Zero
Let
\[ a_n=2n+5,\qquad n=0,1,2,3. \]List the ordered pairs.
Practice 6: Missing Table Value
The following table represents an arithmetic sequence:
| \(n\) | \(a_n\) |
|---|---|
| \(1\) | \(4\) |
| \(2\) | \(9\) |
| \(3\) | \(\square\) |
| \(4\) | \(19\) |
Find the missing value.
Practice 7: Recursive Table
Let
\[ a_1=3,\qquad a_n=2a_{n-1}. \]Create the first five table entries.
Practice 8: Discrete or Continuous?
A sequence is defined by
\[ a_n=n^2,\qquad n=1,2,3,\ldots \]Should the graph automatically include the point
\[ (2.5,6.25)? \]Explain.
Practice 9: Read a Pattern from Points
Consider
\[ (1,7),\ (2,11),\ (3,15),\ (4,19). \]Find the common difference and an explicit rule.
Practice 10: Identify the Error
A student graphs
\[ a_n=3n,\qquad n=1,2,3,4 \]by drawing the entire line
\[ y=3x. \]Explain what is wrong with this as a graph of the sequence itself.
Practice Answers
Practice 1
| \(n\) | \(a_n\) |
|---|---|
| \(1\) | \(4\) |
| \(2\) | \(7\) |
| \(3\) | \(10\) |
| \(4\) | \(13\) |
Practice 2
\[ \boxed{ (0,6),\ (1,9),\ (2,12),\ (3,15) }. \]Practice 3
The sequence is
\[ 4,\ 8,\ 12,\ 16. \]The common difference is \(4\), so
\[ \boxed{a_n=4n}. \]Practice 4
The differences are not constant, but the ratios are:
\[ \frac{10}{5}=2, \qquad \frac{20}{10}=2, \qquad \frac{40}{20}=2. \]Therefore, the sequence is
\[ \boxed{\text{geometric}}. \]Practice 5
\[ a_0=5,\qquad a_1=7,\qquad a_2=9,\qquad a_3=11. \]Therefore,
\[ \boxed{ (0,5),\ (1,7),\ (2,9),\ (3,11) }. \]Practice 6
The common difference is
\[ 9-4=5. \]Therefore,
\[ a_3=9+5=\boxed{14}. \]Practice 7
\[ a_1=3, \] \[ a_2=6, \] \[ a_3=12, \] \[ a_4=24, \] \[ a_5=48. \]| \(n\) | \(a_n\) |
|---|---|
| \(1\) | \(3\) |
| \(2\) | \(6\) |
| \(3\) | \(12\) |
| \(4\) | \(24\) |
| \(5\) | \(48\) |
Practice 8
No.
The sequence is defined only at positive integer indices.
Since
\[ 2.5 \]is not an allowed index,
\[ \boxed{(2.5,6.25)} \]is not automatically part of the sequence graph.
Practice 9
The vertical change is
\[ 11-7=4. \]Thus,
\[ d=4. \]Since \(a_1=7\),
\[ a_n=7+4(n-1) \] \[ =\boxed{4n+3}. \]Practice 10
The sequence is defined only for
\[ n=1,2,3,4. \]Therefore, its graph consists only of
\[ (1,3),\ (2,6),\ (3,9),\ (4,12). \]The entire line \(y=3x\) includes infinitely many additional real-input points that are not members of the sequence.
What to Remember
A sequence can be represented by indexed input-output pairs:
\[ \boxed{(n,a_n)}. \]- A sequence table pairs each allowed index with one term value.
- Each table row corresponds to one ordered pair.
- The index belongs on the horizontal axis.
- The term value belongs on the vertical axis.
- Sequence graphs are normally discrete.
- Arithmetic sequences produce collinear discrete points.
- Geometric sequences produce points on exponential curves.
- Polynomial sequences produce points on corresponding polynomial curves.
- Tables can reveal differences, ratios, recurrences, and missing values.
- All valid representations of the same sequence must agree.
The central translation is
\[ \boxed{ \text{index and term} \longleftrightarrow \text{table row} \longleftrightarrow \text{ordered pair} \longleftrightarrow \text{discrete point} }. \]Where We Go Next
Tables and discrete graphs organize individual sequence terms. We now turn to notation designed to compactly represent repeated operations performed across many indexed terms.
The next lesson, Sigma and Product Notation, introduces compact symbols for repeated addition and multiplication and connects indexed expressions with finite sums and products.