Sequences and Series: Cheat Sheet

Essential Concepts

A sequence is a list of numbers, called terms, written in a specific order.
Explicit formulas define each term of a sequence using the position of the term.
An explicit formula for the [latex]n\text{th}[/latex] term of a sequence can be written by analyzing the pattern of several terms.
Recursive formulas define each term of a sequence using previous terms.
Recursive formulas must state the initial term, or terms, of a sequence.
A set of terms can be written by using a recursive formula.
A factorial is a mathematical operation that can be defined recursively.
The factorial of [latex]n[/latex] is the product of all integers from 1 to [latex]n[/latex]

An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant.
The constant between two consecutive terms is called the common difference.
The common difference is the number added to any one term of an arithmetic sequence that generates the subsequent term.
The terms of an arithmetic sequence can be found by beginning with the initial term and adding the common difference repeatedly.
A recursive formula for an arithmetic sequence with common difference [latex]d[/latex] is given by [latex]{a}_{n}={a}_{n - 1}+d,n\ge 2[/latex].
As with any recursive formula, the initial term of the sequence must be given.
An explicit formula for an arithmetic sequence with common difference [latex]d[/latex] is given by [latex]{a}_{n}={a}_{1}+d\left(n - 1\right)[/latex].
An explicit formula can be used to find the number of terms in a sequence.
In application problems, we sometimes alter the explicit formula slightly to [latex]{a}_{n}={a}_{0}+dn[/latex].

A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant.
The constant ratio between two consecutive terms is called the common ratio.
The common ratio can be found by dividing any term in the sequence by the previous term.
The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly.
A recursive formula for a geometric sequence with common ratio [latex]r[/latex] is given by [latex]{a}_{n}=r{a}_{n - 1}[/latex] for [latex]n\ge 2[/latex] .
As with any recursive formula, the initial term of the sequence must be given.
An explicit formula for a geometric sequence with common ratio [latex]r[/latex] is given by [latex]{a}_{n}={a}_{1}{r}^{n - 1}[/latex].
In application problems, we sometimes alter the explicit formula slightly to [latex]{a}_{n}={a}_{0}{r}^{n}[/latex].

The sum of the terms in a sequence is called a series.
A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum.
The sum of the terms in an arithmetic sequence is called an arithmetic series.
The sum of the first [latex]n[/latex] terms of an arithmetic series can be found using a formula.
The sum of the terms in a geometric sequence is called a geometric series.
The sum of the first [latex]n[/latex] terms of a geometric series can be found using a formula.
The sum of an infinite series exists if the series is geometric with [latex]-1
If the sum of an infinite series exists, it can be found using a formula.
An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series.

Formula for a factorial	[latex]\begin{align}0!&=1\\ 1!&=1\\ n!&=n\left(n - 1\right)\left(n - 2\right)\cdots \left(2\right)\left(1\right)\text{, for }n\ge 2\end{align}[/latex]
recursive formula for nth term of an arithmetic sequence	[latex]{a}_{n}={a}_{n - 1}+d \text{ for } n\ge 2[/latex]
explicit formula for nth term of an arithmetic sequence	[latex]{a}_{n}={a}_{1}+d\left(n - 1\right)[/latex]
recursive formula for [latex]nth[/latex] term of a geometric sequence	[latex]{a}_{n}=r{a}_{n - 1},n\ge 2[/latex]
explicit formula for [latex]nth[/latex] term of a geometric sequence	[latex]{a}_{n}={a}_{1}{r}^{n - 1}[/latex]
sum of the first [latex]n[/latex] terms of an arithmetic series	[latex]{S}_{n}=\dfrac{n\left({a}_{1}+{a}_{n}\right)}{2}[/latex]
sum of the first [latex]n[/latex] terms of a geometric series	[latex]{S}_{n}=\dfrac{{a}_{1}\left(1-{r}^{n}\right)}{1-r} , r\ne 1[/latex]
sum of an infinite geometric series with [latex]-1	[latex]{S}_{n}=\dfrac{{a}_{1}}{1-r}[/latex]

annuity: an investment in which the purchaser makes a sequence of periodic, equal payments

arithmetic sequence: a sequence in which the difference between any two consecutive terms is a constant

common difference: the difference between any two consecutive terms in an arithmetic sequence

common ratio: the ratio between any two consecutive terms in a geometric sequence

explicit formula: a formula that defines each term of a sequence in terms of its position in the sequence

finite sequence: a function whose domain consists of a finite subset of the positive integers [latex]\left\{1,2,\dots n\right\}[/latex] for some positive integer [latex]n[/latex]

geometric sequence: a sequence in which the ratio of a term to a previous term is a constant


index of summation: in summation notation, the variable used in the explicit formula for the terms of a series and written below the sigma with the lower limit of summation

infinite sequence
a function whose domain is the set of positive integers

lower limit of summation: the number used in the explicit formula to find the first term in a series

[latex]n[/latex] factorial: the product of all the positive integers from 1 to [latex]n[/latex]

[latex]n[/latex]th partial sum: the sum of the first [latex]n[/latex] terms of a sequence

[latex]n[/latex]th term of a sequence: a formula for the general term of a sequence

recursive formula: a formula that defines each term of a sequence using previous term(s)

summation notation: a notation for series using the Greek letter sigma; it includes an explicit formula and specifies the first and last terms in the series

upper limit of summation: the number used in the explicit formula to find the last term in a series