Question

1) resolva os fatoriais:
a) (n+2)! / (n+3)! = 20

b) (n+1)!2 /n! =10

2) simplifique os fatoriais:
a) n! / (n - 2)! =

b) (n+1)! / (n+2)1 =

c) (n+3)!(n-1)! / (n-1)!(n+2)! =

3- Resolva os fatorials abaixo:
a) 5! =
b) 4! X 5! =
c) 3!+ 4! =
d) 3! X 5! - 4! =
e) 2! +3! X 2! =
f) 3! / 2!=
G) 3! + 2! / 2! =
h) 3! × 2! / 2! =



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Answer 1

Fatorial de um número natural

É uma multiplicação de fatores até chegar o número 1.

exemplo

$\mathsf{3!=3\cdot2\cdot1=6}$

Um fatorial pode ser escrito em escrito em termos de fatoriais menores.

$\mathsf{n!=n(n-1)!=n(n-1)(n-2)!...3\cdot2\cdot1}$

Definições especiais

$\boxed{\boxed{\boxed{\boxed{\mathsf{1!=1}}}}}\\\huge\boxed{\boxed{\boxed{\boxed{\mathsf{0!=1}}}}}$

$\dotfill$

1) resolva os fatoriais:

a)

$\mathsf{\dfrac{(n+2)!}{(n+3)!}=20}\\\mathsf{\dfrac{(n+2)!}{(n+3)(n+2)!}=20}\\\mathsf{\dfrac{1}{n+3}=20}\\\mathsf{20n+60=1}\\\mathsf{20n=1-60}\\\mathsf{20n=-59}\\\mathsf{n=-\dfrac{59}{20}}\\\huge\boxed{\boxed{\boxed{\boxed{\mathsf{S=\varnothing}}}}}$

b) $\mathsf{\dfrac{2(n+1)!}{n!}=10}\\\mathsf{\dfrac{2(n+1)\cdot n!}{n!}=10}\\\mathsf{n+1=\dfrac{10}{2}}\\\mathsf{n+1=5}\\\mathsf{n=5-1}\\\huge\boxed{\boxed{\boxed{\boxed{\mathsf{n=4}}}}}$

$\dotfill$

2) simplifique os fatoriais:

a)

$\mathsf{\dfrac{n!}{(n-2)!}=\dfrac{n(n-1)(n-2)!}{(n-2)!}=n(n-1)}$

b)

$\mathsf{\dfrac{(n+1)!}{(n+2)!}=\dfrac{(n+1)!}{(n+2)(n+1)!}=\dfrac{1}{n+2}}$

c)

$\mathsf{\dfrac{(n+3)! \cdot(n-1)!}{(n-1)! \cdot(n+2)!}=\dfrac{(n+3)(n+2)!}{(n+2)!}=n+3}$

$\dotfill$

3- Resolva os fatorials abaixo:

a)

$\mathsf{5!=120}$

b)

$\mathsf{4!\cdot5!=24\cdot120=2880}$

c)

$\mathsf{3!+4!=6+24=30}$

d)

$\mathsf{3!\cdot5!-4!=6\cdot120-24=720-24=696}$

e)

$\mathsf{2!+3!\cdot2!=2+6\cdot2=2+12=14}$

f)

$\mathsf{\dfrac{3!}{2!}=\dfrac{3\cdot2!}{2!}=3}$

G)

$\mathsf{\dfrac{3!+2!}{2!}=\dfrac{3\cdot2!+2!}{2!}}\\\mathsf{\dfrac{2!(3+1)}{2!}=4}$

h)

$\mathsf{\dfrac{3!\cdot2!}{2!}=3!=6}$

$\dotfill$

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