Question

The population of a local species of bees can be found using an infinite geometric series where a1=860 and the common ratio is 1/5. Write the sum in sigma notation, and calculate the sum(if possible)that will be the upper limit of this population

Answer 1

$a_1=860$
$a_2=\dfrac{a_1}5$
$a_3=\dfrac{a_2}5=\dfrac{a_1}{5^2}$
$\vdots$
$a_n=\dfrac{a_{n-1}}5=\dfrac{a_{n-2}}{5^2}=\cdots=\dfrac{a_1}{5^{n-1}}$

The $k$th partial sum is

$\displaystyle S_k=\sum_{n=1}^ka_n=\sum_{n=1}^k\frac{a_1}{5^{n-1}}$
$S_k=a_1\left(1+\dfrac15+\dfrac1{5^2}+\cdots+\dfrac1{5^{k-2}}+\dfrac1{5^{k-1}}$
$\dfrac15S_k=a_1\left(\dfrac15+\dfrac1{5^2}+\dfrac1{5^3}+\cdots+\dfrac1{5^{k-1}}+\dfrac1{5^k}\right)$

$\implies S_k-\dfrac15S_k=a_1\left(1-\dfrac1{5^k}\right)$
$\dfrac45 S_k=860\left(1-\dfrac1{5^k}\right)$
$S_k=1075-\dfrac{1075}{5^k}$

As $k\to\infty$, we're left with

$\displaystyle\sum_{n=1}^\infty a_n=\lim_{k\to\infty}\left(1075-\frac{1075}{5^k}\right)=1075$

which is the upper limit to the population of the bees.