Question

The population of a local species of dragon fly can be found using an infinite geometric series where a1=36 and the common ratio is 1/2 write the sum in sigma notation and calculate the sum that will be the upper limit of this population

Answer 1

Hello,

$a_0=72\\ a_1=72* \dfrac{1}{2}=36 \\ ...\\ a_i=72* \dfrac{1}{2^i} \\\\ \lim_{n \to \infty} \sum_{i=0}^{\infty}\ a_i = \lim_{n \to \infty}72* \sum_{i=0}^{\infty}\ \dfrac{1}{2^i} \\\\ =72* \lim_{n \to \infty} \dfrac{1-\frac{1}{2^{n+1}}}{1-\frac{1}{2}} \\\\ =72*2=144$