Question

The population of a local species of dragonfly can be found using an infinite geometric series where a1 = 36 and the common ratio is one half. Write the sum in sigma notation and calculate the sum that will be the upper limit of this population.

Answer 1

Option 3 -
`S=\sum_(i=1)^(\infty) 36((1)/(2))^i-1` ; the sum is 72.

Explanation:

Given : The population of a local species of dragonfly can be found using an infinite geometric series where
`a_1 = 36` and the common ratio is one half.

To find : Write the sum in sigma notation and calculate the sum that will be the upper limit of this population.

Solution :

Geometric series is
`a+ar+ar^2+ar^3+.......`

The formula of sum of geometric infinite series is

`S=\sum_(k=0)^(\infty) ar^k=(a)/(1-r)`

In the given geometric series,

`a_1 = 36` and
`r=(1)/(2)`

According to question,

`S=\sum_(i=1)^(\infty) 36((1)/(2))^i-1=(36)/(1-(1)/(2))`

`S=\sum_(i=1)^(\infty) 36((1)/(2))^i-1=(36)/((1)/(2))`

`S=\sum_(i=1)^(\infty) 36((1)/(2))^i-1=72`

Therefore, Option 3 is correct.

`S=\sum_(i=1)^(\infty) 36((1)/(2))^i-1` ; the sum is 72.

Answer 2

The third option is the correct answer.