Question

The population of a local species of dragonfly can be found using an infinite geometric series where a1=42 and the common ratio is 3/4. Write the sum in sigma notation and calculate the sound that will be the upper limit of this population

Answer 1

$\bf \textit{sum of an infinite geometric serie}\\\\ \stackrel{for~~|r|\ \textless \ 1}{S=\sum\limits_{i=0}^{\infty}~a_1r^i\implies \cfrac{a_1}{1-r}}\qquad \begin{cases} a_1=\textit{first term's value}\\ r=\textit{common ratio}\\ ----------\\ a_1=42\\ r=\frac{3}{4} \end{cases} \\\\\\ S=\cfrac{42}{1-\frac{3}{4}}\implies S=\cfrac{42}{\frac{1}{4}}\implies S=164$

bearing in mind that, the geometric sequence is "convergent" only when |r|<1, or namely "r" is a fraction between 0 and 1.