Question

The population of a local species of beetle can be found using an infinite geometric series where a1 = 880 and the common ratio is one forth. Write the sum in sigma notation and calculate the sum (if possible) that will be the upper limit of this population.

Answer 1

Given : The population of a local species of beetle can be found using an infinite geometric series where a1 = 880 and the common ratio is one forth.

To Find: Write the sum in sigma notation and calculate the sum (if possible) that will be the upper limit of this population.

Solution:

`a_1=880`

`r = (1)/(4)`

Since we are given that it is an infinite geometric series

So, to find the sum in sigma notation and calculate the sum (if possible) that will be the upper limit of this population.

`\sum a_n=(a_1)/(1-r)`

And we need to calculate the sum that will be the upper limit of this population:

`\sum a_n=(880)/(1-(1)/(4))=(880)/((3)/(4))=(880* 4)/(3)=1173.3`

Hence the sigma notation is
`\sum a_n=(a_1)/(1-r)` and the sum is 1173.3.

Answer 2

Using the formula:
A1/(1 - r)

Substituting the values:

= 880 /(1 - 1/4)
= 880 / (3/4)
= 1173.33