What Does! Mean in Sequences?
$\Begingroup$ I'm Doing a Sequences Problem Where I Have to Write the First Five Terms of a Sequence. It Looks Normal, but There Is an Exclamation Mark on the...
I'm doing a sequences problem where I have to write the first five terms of a sequence. It looks normal, but there is an exclamation mark on the denominator:
$$a_n = \frac{1}{(n + 1)!}$$
&
$$a_n = \frac{(-1)^{n}n}{n! + 1}$$
What does the exclamation mark mean, and how do I go about this differently with the exclamation mark?
The factorial of a number is represented by the exclamation point (!). The factorial of a number $x$ is often described as the product of all positive integers less then or equal to $x$. For example:
$$4! = 4\cdot3\cdot2\cdot1 = 24$$
It is often also useful to describe $x!$ in a recursive relation:
$$x! = x(x-1)!$$
where $0! = 1$.
This method is often good because it helps explain why $0!=1$ (see also the "empty product" for more on this). Using the above explanation, we may find $4!$:
$$4! = 4(3)! = 4(3)(2)! = \cdots = 4(3)(2)(1)(0)! = 4(3)(2)(1)(1) = 24$$
If the number is 4! then you count down so it would be $4\times 3 \times 2\times 1= 24$. Another thing is that if the number is 7! you would count down until you get to two. So it would be $7\times 6\times 5\times 3\times 2 =5040$ 7 x 6 x 5 x 4 x 3 x 2 I just put in the 4 that was not listed although the answer is still 5040.
Chloe Bennett explores the intersection of pop culture, streaming entertainment, digital trends, and contemporary lifestyle. Her weekly commentary reaches thousands of culture enthusiasts.
