The formulas for some sequences include products of consecutive positive integers.
$n$ factorial , written as
$n!,$ is the product of the positive integers from 1 to
$n.$ For example,
The factorial of any whole number
$n$ is
$n(n-1)!$ We can therefore also think of
$5!$ as
$\text{\hspace{0.17em}}5\cdot 4!\text{.}$
Factorial
n factorial is a mathematical operation that can be defined using a recursive formula. The factorial of
$\text{\hspace{0.17em}}n,\text{\hspace{0.17em}}$ denoted
$\text{\hspace{0.17em}}n!,\text{\hspace{0.17em}}$ is defined for a positive integer
$n$ as:
Can factorials always be found using a calculator?
No. Factorials get large very quickly—faster than even exponential functions! When the output gets too large for the calculator, it will not be able to calculate the factorial.
Writing the terms of a sequence using factorials
Write the first five terms of the sequence defined by the explicit formula
${a}_{n}=\frac{5n}{(n+2)!}.$
Substitute
$\text{\hspace{0.17em}}n=1,n=2,\text{\hspace{0.17em}}$ and so on in the formula.
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. See
[link] ,
[link] , and
[link] .
An explicit formula for the
$\text{\hspace{0.17em}}n\text{th}\text{\hspace{0.17em}}$ term of a sequence can be written by analyzing the pattern of several terms. See
[link] .
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. See
[link] and
[link] .
A factorial is a mathematical operation that can be defined recursively.
The factorial of
$\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is the product of all integers from 1 to
$\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ See
[link] .
Questions & Answers
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5) and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.