Question

a soma dos 20 termos de uma p.a. infinita é igual a 710. se o primeiro termo dessa p.a. é 01 igual a 7, calcule seu 10 termo

Answer 1

$\mathrm{PA=(a_1;a_2;a_3;\dots;a_{10};\dots;a_{20};\dots)\ \to\ a_1=7}\\\\ \mathbf{Pelo\ termo\ geral,\ obteremos:}\\\\ \mathrm{a_n=a_1+(n-1)r\ \to\ a_{20}=a_1+19r\ \to\ \boxed{\mathrm{a_{20}=7+19r}}}\\\\ \mathbf{Pela\ soma\ dos\ termos,\ obteremos:}\\\\ \mathrm{S_{n}=(a_1+a_n)\dfrac{n}{2}\ \to\ S_{20}=(a_1+a_{20})\dfrac{20}{2}\ \to\ 710=(7+a_{20})10}\\\\ \mathrm{7+a_{20}=71\ \to\ a_{20}=64\ \to\ 7+19r=64\ \to\ 19r=57\ \to\ \boxed{\mathrm{r=3}}}\\\\ \mathbf{Calculando\ a_{10}:}\\\\ \mathrm{a_{10}=a_1+9r=7+9.3=7+27=34\ \to\ \boxed{\mathbf{a_{10}=34}}}$