Question

Calcule a soma dos múltiplos de 5 compreendidos entre 42 e 448.

Answer 1

$\mathbf{Trataremos\ os\ m\acute{u}ltiplos\ de\ 5\ entre\ 42\ e\ 448}\\ \mathbf{como\ uma\ progress\~ao\ aritm\acute{e}tica\ (PA):}\\\\ \mathrm{\Rightarrow O\ 1\°\ m\acute{u}ltiplo\ \acute{e}\ o\ 45\ \to\ \boxed{\mathrm{a_1=45}}}\\\\ \mathrm{\Rightarrow O\ \acute{u}ltimo\ m\acute{u}ltiplo\ \acute{e}\ o\ 445\ \to\ \boxed{\mathrm{a_n=445}}}\\\\ \mathrm{\Rightarrow Raz\~ao\ \to\ \boxed{\mathrm{r=5}}}$

$\mathbf{Pelo\ termo\ geral\ de\ uma\ PA,\ teremos\ que:}\\\\ \mathrm{a_n=a_1+(n-1)r\ \to\ 445=45+(n-1).5\ \to}\\\\ \mathrm{\to\ (n-1).5=445-45\ \to\ n-1=\dfrac{400}{5}\ \to}\\\\ \mathrm{\to\ n-1=80\ \to\ n=80+1\ \to\ \boxed{\mathrm{n=81}}}$

$\mathbf{Pela\ soma\ dos\ termos\ de\ uma\ PA,\ teremos\ que:}\\\\ \mathrm{S_n=\dfrac{n}{2}(a_1+a_n)\ \to\ S_{81}=\dfrac{81}{2}(45+445)=}\\\\ \mathrm{=\dfrac{81}{2}490=81.245=19845\ \to\ \boxed{\mathbf{S=19845}}}$