Questão 7103

FGV 1975

Matemática

(FGV - 1975) O determinante é igual a:

Gabarito:

Resolução:

1) Sabendo que $inom{m}{r} = frac{m!}{r!(m-r)!}$

2) Com isso, temos que:

$egin{vmatrix} 1 & inom{n}{1} & inom{n+1}{1} \ 1 & inom{n+1}{1} & inom{n+2}{1} \ 1 & inom{n+2}{1} & inom{n+3}{1}end{vmatrix} = egin{vmatrix} 1 & frac{n!}{(n-1)!} & frac{(n+1)!}{n!} \ 1 & frac{(n+1)!}{n!} & frac{(n+2)!}{(n+1)!} \ 1 & frac{(n+2)!}{(n+1)!} & frac{(n+3)!}{(n+2)!} end{vmatrix}$

3) Aplicando a Regra de Chio:

$egin{vmatrix} 1 & inom{n}{1} & inom{n+1}{1} \ 1 & inom{n+1}{1} & inom{n+2}{1} \ 1 & inom{n+2}{1} & inom{n+3}{1}end{vmatrix} = egin{vmatrix} frac{(n+1)!}{n!}- frac{n!}{(n-1)!} & frac{(n+2)!}{(n+1)!}- frac{(n+1)!}{n!} \ frac{(n+2)!}{(n+1)!}- frac{n!}{(n-1)!} & frac{(n+3)!}{(n+2)!} - frac{(n+1)!}{n!}end{vmatrix}$

4) Desenvolvendo:

$egin{vmatrix} 1 & inom{n}{1} & inom{n+1}{1} \ 1 & inom{n+1}{1} & inom{n+2}{1} \ 1 & inom{n+2}{1} & inom{n+3}{1}end{vmatrix} = egin{vmatrix} 1 & 1 \ 2 & 2end{vmatrix}=0$

Questão 7103

FGV 1975

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