Question

What is the sum of an infinite geometric series if a1 = 144 and r = 1⁄4?

A. 192
B. 288
C. 576
D. 252

Answer 1

`\bf \qquad \qquad \textit{sum of a infinite geometric sequence} \\\\ S=\sum\limits_(i=0)^(\infty)\ a_1\cdot r^(i)\implies S=\cfrac{a_1}{1-r}~~ \begin{cases} a_1=\textit{first term's value}\\ r=\textit{common ratio}\\ \qquad 0\leqslant |r| \leqslant 1\\[-0.5em] \hrulefill\\ r=(1)/(4)\\ a_1=144 \end{cases} \\\\\\ S=\cfrac{144}{1-(1)/(4)}\implies S=\cfrac{144}{(3)/(4)}\implies S=192`

bearing in mind that 0⩽|r|⩽1, is just another way to say "r" is a proper fraction, and in this case, it's 1/4, so it's.