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

The third option is the correct answer.

Answer 2

Answer:

Option 3 - $S=\sum_{i=1}^{\infty} 36(\frac{1}{2})^i-1$ ; the sum is 72.

Step-by-step 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

$S=\sum_{k=0}^{\infty} ar^k=\frac{a}{1-r}$

In the given geometric series,

$a_1 = 36$ and $r=\frac{1}{2}$

According to question,

$S=\sum_{i=1}^{\infty} 36(\frac{1}{2})^i-1=\frac{36}{1-\frac{1}{2}}$

$S=\sum_{i=1}^{\infty} 36(\frac{1}{2})^i-1=\frac{36}{\frac{1}{2}}$

$S=\sum_{i=1}^{\infty} 36(\frac{1}{2})^i-1=72$

Therefore, Option 3 is correct.

$S=\sum_{i=1}^{\infty} 36(\frac{1}{2})^i-1$ ; the sum is 72.