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=\frac{a_1}5`

`a_3=\frac{a_2}5=(a_1)/(5^2)`

`\vdots`

`a_n=\frac{a_(n-1)}5=(a_(n-2))/(5^2)=\cdots=(a_1)/(5^(n-1))`

The
`k`th partial sum is


`\displaystyle S_k=\sum_(n=1)^ka_n=\sum_(n=1)^k(a_1)/(5^(n-1))`

`S_k=a_1\left(1+\frac15+\frac1{5^2}+\cdots+\frac1{5^(k-2)}+\frac1{5^(k-1)}`

`\frac15S_k=a_1\left(\frac15+\frac1{5^2}+\frac1{5^3}+\cdots+\frac1{5^(k-1)}+\frac1{5^k}\right)`


`\implies S_k-\frac15S_k=a_1\left(1-\frac1{5^k}\right)`

`\frac45 S_k=860\left(1-\frac1{5^k}\right)`

`S_k=1075-(1075)/(5^k)`

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


`\displaystyle\sum_(n=1)^\infty a_n=\lim_(k\to\infty)\left(1075-(1075)/(5^k)\right)=1075`

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