Question

What is yhe sum of an infinite geometric series if a1=5 and r=1/3

Answer 1

bearing in mind that an infinite geometric sequence, has a limit, namely converges at a value, only if "r" the common factor, is a proper fraction, namely | r | < 1, in this case it's so, thus

$\bf \qquad \qquad \textit{sum of an infinite geometric sequence} \\\\ S=\sum\limits_{i=0}^{\infty}\ a_1 r^{i}\implies S=\cfrac{a_1}{1-r}\quad \begin{cases}a_1=\textit{first term's value}\\ r=\textit{common ratio}\\[-0.5em] \hrulefill\\ a_1=5\\ r=\frac{1}{3} \end{cases} \\\\\\ S=\cfrac{~~5~~}{1-\frac{1}{3}}\implies S=\cfrac{~~5~~}{\frac{2}{3}}\implies S=\cfrac{~~\frac{5}{1}~~}{\frac{2}{3}}\implies S=\cfrac{15}{2}$