Question

calcule a soma dos termos da pg (10,2,...)

Answer 1

Dada a P.G (Progressão Geometrica):

$\mathsf{\left(10,2,...\right)}$

= = = = =

Calculando a razão (q) da P.G:

$\mathsf{r=\dfrac{a_n}{a_{n-1}}}\\\\\\\mathsf{r=\dfrac{a_2}{a_1}}\\\\\\\mathsf{r=\dfrac{2}{10}}\\\\\\\mathsf{r=\dfrac{1}{5}}$

= = = = =

A soma de uma P.G infinita é determinada pela formula:

$\mathsf{S_n=\dfrac{a_1}{1-q}}\\\\\\\mathsf{S_n=\dfrac{10}{1-\frac{1}{5}}}\\\\\\\mathsf{S_n=\dfrac{10}{\frac{5}{5}-\frac{1}{5}}}\\\\\\\mathsf{S_n=\dfrac{10}{\frac{5-1}{5}}}\\\\\\\mathsf{S_n=\dfrac{10}{\frac{4}{5}}}\\\\\\\mathsf{S_n=10\cdot \dfrac{5}{4}}\\\\\\\mathsf{S_n=\dfrac{50}{4}}\\\\\\\mathsf{S_n=\dfrac{25}{2}}\\\\\\\boxed{\mathsf{S_n=12,5}}\: \: \checkmark$