Question

Sabendo que f(x+1) = 3x2 + f(x) e que f(0) =0 , determine o valor de f(1) + f(2) + f(3)

Answer 1

Resposta:

$\boxed{\mathtt{18}}$

Explicação passo-a-passo:

De acordo com o enunciado, $\displaystyle \mathtt{f(x + 1) = 3x^2 + f(x)}$ e $\displaystyle \mathtt{f(0) = 0}$. Determinemos $\displaystyle \mathtt{f(1)}$ fazendo $\displaystyle \mathtt{x = 0}$...

$\\ \displaystyle \mathsf{f(x + 1) = 3x^2 + f(x)} \\\\ \mathsf{f(0 + 1) = 3 \cdot 0^2 + f(0)} \\\\ \mathsf{f(1) = 0 + 0} \\\\ \boxed{\mathsf{f(1) = 0}}$

Encontremos $\displaystyle \mathtt{f(2)}$ tomando $\displaystyle \mathtt{x = 1}$...

$\\ \displaystyle \mathsf{f(x + 1) = 3x^2 + f(x)} \\\\ \mathsf{f(1 + 1) = 3 \cdot 1^2 + f(1)} \\\\ \mathsf{f(2) = 3 + 0} \\\\ \boxed{\mathsf{f(2) = 3}}$

Para determinar $\displaystyle \mathtt{f(3)}$ fazemos $\displaystyle \mathtt{x = 2}$. Segue,

$\\ \displaystyle \mathsf{f(x + 1) = 3x^2 + f(x)} \\\\ \mathsf{f(2 + 1) = 3 \cdot 2^2 + f(2)} \\\\ \mathsf{f(3) = 3 \cdot 4 + 3} \\\\ \boxed{\mathsf{f(3) = 15}}$

Por fim, temos que:

$\\ \displaystyle \mathsf{f(1) + f(2) + f(3) = 0 + 3 + 15} \\\\ \boxed{\boxed{\mathsf{f(1) + f(2) + f(3) = 18}}}$