Question

Determine a soma dos termos das seguintes progressões geometricas infinitas: A) (10, 4, 8/5,...) C) (100,-10,1,...)

Answer 1

a)

$PG=(10;4;\frac{8}{5};...)\\\\ q=\frac{a_{n}}{a_{n-1}}\ \to\ q=\frac{a_2}{a_1}\\\\ q=\frac{4}{10}\ \to\ \| \ q=\frac{2}{5}\ \| \\\\ S_\infty=\frac{a_1}{1-q}\ \to\ S_\infty= \frac{10}{1-\frac{2}{5}}\\\\ S_\infty=\frac{10}{\frac{5-2}{5}}\ \to\ S_\infty=\frac{10}{\frac{3}{5}}\\\\ S_\infty=10.\frac{5}{3}\ \to\ \| \ S_\infty=\frac{50}{3}\ \|$

b)

$PG=(100;-10;1;...)\\\\ q=\frac{a_n}{a_{n-1}}\ \to\ q=\frac{a_2}{a_1}\\\\ q=\frac{-10}{100}\ \to\ \| \ q=\frac{-1}{10}\ \|\\\\ S_\infty=\frac{a_1}{1-q}\ \to\ S_\infty=\frac{100}{1-(\frac{-1}{10})}\\\\ S_\infty=\frac{100}{\frac{10+1}{10}}\ \to\ S_\infty=\frac{100}{\frac{11}{10}}\\\\ S_\infty=100.\frac{10}{11}\ \to\ \| \ S_\infty=\frac{1000}{11}\ \|$