2021-12-26

What is the factorial of $\left(n+1\right)$ ?

zurilomk4

It is $n!\cdot \left(n+1\right)$
Explanation:
Since factorial n (or n!)) is the product of all numbers up to and including n, we only have to multiply by the next number.

Virginia Palmer

As we know $n\ne n×\left(n-1\right)!$
Put $n=n+1$ in above equation, hence
$\left(n+1\right)\ne \left(n+1\right)×\left[\left(n+1\right)-1\right]!$
$\left(n+1\right)=\left(n×1\right)×\left[n+1-1\right]!$
$\left(n+1\right)\ne \left(n+1\right)×n$
Hence proved.

nick1337

What is the factorial of n?
If n is some positive integer, then the factorial of n is the product of every natural number till n, or
$n!=1\cdot 2\cdot 3\cdot ...\cdot n$
And that way, the factorial of n+1 becomes
$\left(n+1\right)!=1\cdot 2\cdot 3\cdot ...\cdot n\cdot \left(n+1\right)$
As you can clearly observe, the part of the second expansion till n is equal to the first expansion, the one for n factorial.
So,
$\left(n+1\right)!=n!\cdot \left(n+1\right)$

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