Question

Determine o número inteiro positivo que verifica a equação An,3 = 3 . (n - 1)

Answer 1

Explicação passo-a-passo:

$\sf A_{n,3}=3\cdot(n-1)$

$\sf \dfrac{n!}{(n-3)!}=3\cdot(n-1)$

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

$\sf n\cdot(n-1)\cdot(n-2)=3\cdot(n-1)$

$\sf n\cdot(n-2)=3$

$\sf n^2-2n-3=0$

$\sf \Delta=(-2)^2-4\cdot1\cdot(-3)$

$\sf \Delta=4+12$

$\sf \Delta=16$

$\sf n=\dfrac{-(-2)\pm\sqrt{16}}{2\cdot1}=\dfrac{2\pm4}{2}$

$\sf n'=\dfrac{2+4}{2}~\longrightarrow~n'=\dfrac{6}{2}~\longrightarrow~n'=3$

$\sf n"=\dfrac{2-4}{2}~\longrightarrow~n"=\dfrac{-2}{2}~\longrightarrow~n'=-1$ (não serve)

$\sf S=\{3\}$