Question

Resolva a seguinte equação:

( n + 2 )! - ( n + 1 )!/ n( n - 1 )! = 25

Answer 1

Olá !

Basta desenvolver os fatoriais e resolver ...

(n + 2)! - ( n + 1)! /n.(n - 1)! = 25

desenvolvendo ...

[(n+2).(n+1).n.(n-1)! - (n+1).n.(n-1)!]/n.(n-1)!  = 25

corto todos os (n-1)! ...

[(n+2).(n+1).n - (n+1).n]/n = 25

corto todos os n ...

(n+2).(n+1) - (n+1) = 25  

n² + n + 2n + 2 - n - 1 = 25

n² + 2n + 1 = 25

n² + 2n - 24 = 0                           Equação do 2º grau

Δ = 4 + 96

Δ = 100

n = (-2 +- √100)/2                          (descarto o - √100)

n = (-2 + 10)/2

n = 8/2

n = 4  

ok

Answer 2

Resposta:

$\red{ \boxed{ \green{ \boxed{ \pink{ \boxed{ \sf{n = 4}} }} } } }$

Explicação passo-a-passo:

$\blue {{ _{\heartsuit} } \heartsuit_{\heartsuit} } \: \: \mathsf {OI, TUDO \: \: J\acute{O}IA \: ? \: \:\blue {{ _{\heartsuit} } \heartsuit_{\heartsuit} } }$

$\sf{ \frac{( n + 2)! - (n + 1)!}{n \times (n - 1)!} = 25} \\ \\ \sf{ \frac{(n + 2) \times (n + 1)! - (n + 1)!}{n \times (n - 1)!} = 25} \\ \\ \sf{ \frac{(n + 2) \times (n + 1)! - (n + 1)!}{(n - 1)! \times n} = 25} \\ \\ \sf{ \frac{(n + 2 - 1) \times (n + 1)!}{(n - 1)! \times n} = 25} \\ \\ \sf{ \frac{(n + 2 - 1) \times (n + 1)!}{n!} = 25 } \\ \\ \sf{ \frac{(n + 1) \times (n + 1)!}{n!} = 25} \\ \\ \sf{ \frac{(n + 1) \times (n + 1) \times \cancel{\blue {n!}}}{\cancel{\blue {n!}}} = 25} \\ \\ \sf{ (n + 1) \times (n + 1) = 25} \\ \\ \sf{ {(n + 1)}^{1} \times {(n + 1)}^{1} = 25} \\ \\ \sf{ {(n + 1)}^{1 + 1} = 25 } \\ \\ \sf{ {(n + 1)}^{2} = 25 } \\ \\ \sf{n + 1 = \pm5} \\ \\ \red{ \sf{n + 1 = - 5}} \\ \green{ \sf{n + 1 = 5}} \\ \\ \red{ \sf{n = - 6}} \\ \green{ \sf{n = 4}} \\ \\ \red{ \sf{ \frac{( - 6 + 2)! - ( - 6 + 1)!}{ - 6 \times ( - 6 - 1)!} = 25}} \\ \green{ \sf{ \frac{(4 + 2)! - (4 + 1)!}{4 \times (4 - 1)!} = 25 }} \\ \\ \green{ \frac{6! - (4 + 1)!}{4 \times (4 - 1)!} = 25 } \\ \\ \green{ \frac{6! - 5!}{4 \times (4 - 1)!} = 25 } \\ \\ \green{ \sf{ \frac{6! - 5!}{4 \times 3!} = 25 }} \\ \\ \green{ \sf{ \frac{6 \times 5! - 5!}{4 \times 3!} = 25 }} \\ \\ \green{ \sf{ \frac{6 \times 5! - 5!}{3! \times 4} = 25}} \\ \\ \green{ \sf{ \frac{5 \times 5!}{3! \times 4} = 25}} \\ \\ \green{ \sf{ \frac{5 \times 5!}{4!} = 25}} \\ \\ \green{ \sf{ \frac{5 \times 5 \times \cancel{\blue {4!}}}{\cancel{\blue {4!}}} = 25}} \\ \\ \green{ \sf{5 \times 5 = 25}} \\ \\ \green{ \sf{25 = 25}} \\ \\ \red{ \sf{ \frac{( - 6 + 2)! - ( - 6 + 1)!}{ - 6 \times ( - 6 - 1)!} = 25}} \\ \\ \red{ \sf{ \frac{( - 4)! - ( - 6 + 1)!}{ - 6 \times ( - 6 - 1)!} = 25}} \\ \\ \red{ \sf{n \neq - 6}} \\ \green{ \sf{25 = 25 }} \\ \\ \red{ \sf{n \neq - 6}} \\ \green{ \sf{n = 4}} \\ \\ \red{ \boxed{ \green{ \boxed{ \pink{ \boxed{ \sf{n = 4}} }} } } }$

$\mathcal{ATT : ARMANDO}$