Sequences

A sequence is a function whose domain is the set of natural numbers. Instead of writing \(a(n)\) for the value of the sequence at \(n\), we generally write \(a_{n}\), and we refer to this value as the \(\boldsymbol{n}\) th term of the sequence. Sequences are often described in list form:

\[ a_{1}, a_{2}, a_{3}, \ldots \]

Partial Sums of a Sequence

For the sequence \(a_{1}, a_{2}, a_{3}, \ldots\) the \(\boldsymbol{n}\) th partial sum \(S_{n}\) is the sum of the first \(n\) terms of the sequence:

\[ S_{n}=a_{1}+a_{2}+a_{3}+\cdots+a_{n} \]

The \(n\)th partial sum of a sequence can also be expressed by using sigma notation:

\[ S_{n}=\sum_{k=1}^{n} a_{k} \]

Arithmetic Sequences

An arithmetic sequence is a sequence whose terms are obtained by adding the same fixed constant \(d\) to each term to get the next term. Thus an arithmetic sequence has the form

\[ a, a+d, a+2 d, a+3 d, \ldots \]

The number \(a\) is the first term of the sequence, and the number \(d\) is the common difference. The \(n\)th term of the sequence is

\[ a_{n}=a+(n-1) d \]

Partial Sums of an Arithmetic Sequence

For the arithmetic sequence \(a_{n}=a+(n-1) d\) the \(n\)th partial \(\operatorname{sum} S_{n}=\sum_{k=1}^{n}[a+(k-1) d]\) is given by either of the following equivalent formulas:

  1. \(S_{n}=\frac{n}{2}[2 a+(n-1) d]\)
  2. \(S_{n}=n\left(\frac{a+a_{n}}{2}\right)\)

Geometric Sequences

A geometric sequence is a sequence whose terms are obtained by multiplying each term by the same fixed constant \(r\) to get the next term. Thus a geometric sequence has the form

\[ a, a r, a r^{2}, a r^{3}, \ldots \]

The number \(a\) is the first term of the sequence, and the number \(r\) is the common ratio. The \(n\)th term of the sequence is

\[ a_{n}=a r^{n-1} \]

Partial Sums of a Geometric Sequence

For the geometric sequence \(a_{n}=a r^{n-1}\) the \(n\)th partial sum \(S_{n}=\sum_{k=1}^{n} a r^{k-1}(\) where \(r \neq 1)\) is given by

\[ S_{n}=a \frac{1-r^{n}}{1-r} \]

Infinite Geometric Series

An infinite geometric series is a series of the form

\[ a+a r+a r^{2}+a r^{3}+\cdots+a r^{n-1}+\cdots \]

An infinite geometric series for which \(|r|<1\) has the sum

\[ S=\frac{a}{1-r} \]

Principle of Mathematical Induction

For each natural number \(n\), let \(P(n)\) be a statement that depends on \(n\). Suppose that each of the following conditions is satisfied.

  1. \(P(1)\) is true.
  2. For every natural number \(k\), if \(P(k)\) is true, then \(P(k+1)\) is true.

Then \(P(n)\) is true for all natural numbers \(n\).