Examples of Sequences

Sequences as Mathematical Patterns

A sequence is a function from \(\mathbb{N}_0\) to \(\mathbb{R}\), and its terms are written \(a_0,a_1,a_2,\ldots\). The notation tells us which value belongs to each index; examples show the different ways a sequence can be formed. Some are given by a formula that can be evaluated at any index, while others are specified by a starting value and a rule for producing each next term.

A list of initial terms can suggest a pattern, but it does not by itself define all later terms. To establish a sequence, we need a rule that assigns a value at every allowed index. In the examples below, explicit formulas and recurrences provide that rule. We will also examine what simple properties can be proved from those definitions, particularly boundedness and periodicity.

These definitions specify how neighboring terms are related. An arithmetic sequence adds the same amount at each step; a geometric sequence multiplies by the same amount. When the first term is \(a_0\), repeated use of the respective rule gives the explicit formulas \(a_n=a_0+nd\) and \(a_n=a_0r^n\). In the geometric formula, the convention \(r^0=1\) applies also when \(r=0\).

Arithmetic Sequences: Adding a Fixed Amount

For an arithmetic sequence, each step changes the term by the common difference \(d\). A positive difference makes the terms increase, a negative difference makes them decrease, and a zero difference makes every term equal. The explicit formula follows by adding \(d\) once for each step from index \(0\) to index \(n\).

Proof. If \(\alpha=0\), then \(a_n=\beta\) for every \(n\), so \(|a_n|=|\beta|\) for every index. The sequence is bounded.

Now suppose \(\alpha\neq0\). The Triangle Inequality, established earlier in this course, gives

Rearranging yields \(|a_n|=|\alpha n+\beta|\geq|\alpha|n-|\beta|\). Given any \(M\in\mathbb{R}\), the Archimedean Property provides a positive integer \(n\) such that

Here the denominator is positive because \(\alpha\neq0\). Multiplying by \(|\alpha|\) and subtracting \(|\beta|\) gives \(|a_n|\geq|\alpha|n-|\beta|>M\). Thus no real number can bound the absolute values of all terms, and the sequence is not bounded. This proves both directions. \(\square\)

The theorem is useful when an arithmetic sequence is given by an explicit linear formula: its slope determines boundedness. For example, \(5-2n\) is unbounded because its coefficient of \(n\) is nonzero. The conclusion concerns boundedness, not whether a particular term is positive; an unbounded sequence may still have many positive terms.

Geometric Sequences: Multiplying by a Fixed Amount

The common ratio controls how the size of a geometric sequence changes. A ratio between \(-1\) and \(1\) in absolute value does not increase the absolute value of a term. A negative ratio also changes the sign at each step when the starting term is nonzero. If the absolute value of the ratio exceeds \(1\), repeated multiplication produces terms with arbitrarily large absolute value, unless the starting term is zero.

Proof. If \(c=0\), then \(a_n=0\) at every index, so the sequence is bounded. Suppose now that \(c\neq0\). If \(|r|\leq1\), then \(0\leq |r|^n\leq1\) for every \(n\in\mathbb{N}_0\). This includes \(r=0\): at \(n=0\), \(r^0=1\), and at every positive index \(r^n=0\). Consequently,

for every \(n\), so the sequence is bounded.

For the other direction, suppose \(|r|>1\), and set \(h=|r|-1\). Then \(h>0\) and \(|r|=1+h\). Bernoulli’s Inequality, established earlier in this course, gives \((1+h)^n\geq1+nh\) for every nonnegative integer \(n\). Therefore

Given any real \(M\), the Archimedean Property allows us to choose a positive integer \(n\) such that \(n>(M/|c|-1)/h\). Since \(|c|h>0\), this choice implies \(|c|(1+nh)>M\). The displayed inequality then gives \(|a_n|>M\). Thus the absolute values of the terms have no real upper bound, so the sequence is not bounded. This proves the theorem. \(\square\)

Other Useful Patterns

Not every sequence is arithmetic or geometric. Some examples use a formula whose behavior changes with the index, while others are most naturally described by a recurrence. The next examples illustrate patterns that are easy to recognize but should still be interpreted through their precise definitions.

A sequence may also repeat a finite pattern. Repetition can be expressed with a single rule that applies at every index, rather than by displaying only a few terms and relying on an ellipsis.

Proof. Let \((a_n)\) be periodic with period \(p\). Each nonnegative integer \(n\) can be written, by the Division Algorithm, as \(n=qp+r\), where \(q\in\mathbb{N}_0\) and \(r\) is an integer satisfying \(0\leq r<p\). Repeated use of \(a_{k+p}=a_k\) gives \(a_{qp+r}=a_r\). More explicitly, applying the period relation \(q\) times reduces the index \(qp+r\) to \(r\); if \(q=0\), the equality is immediate. Thus every term is one of the finitely many values \(a_0,a_1,\ldots,a_{p-1}\). A finite nonempty set of real numbers has a largest absolute value, so let \(C=\max\{|a_0|,\ldots,|a_{p-1}|\}\). Then \(|a_n|=|a_r|\leq C\) for every \(n\), proving that the sequence is bounded. \(\square\)

Reading Examples Carefully

These examples show why the defining rule matters more than the first few displayed values. The arithmetic rule adds a fixed difference, the geometric rule multiplies by a fixed ratio, and a periodic rule repeats after a fixed number of indices. A reciprocal formula or an alternating formula has its own index-by-index meaning. Similar-looking lists can arise from different rules, so a pattern suggested by initial terms should not be treated as a proof about every term.

Boundedness also depends on the family and its parameters. A nonzero slope makes a linear sequence unbounded; for a geometric sequence with nonzero starting value, a ratio of absolute value greater than \(1\) makes it unbounded. By contrast, periodicity guarantees boundedness because only finitely many values can occur. These are conclusions proved from the definitions, not guesses based on how the first few terms appear.