Question

The population of a local species of dragonfly can be found using an infinite geometric series where a1 = 42 and the common ratio is 3/4. Write the sum in sigma notation and calculate the sum that will be the upper limit of this population.

Answer 1

we are given

first term is

$a_1=42$

common ratio is

$r=\frac{3}{4}$

now, we can find nth term

$a_i=a_1(r)^{i-1}$

now, we can plug values

$a_i=42(\frac{3}{4})^{i-1}$

now, we can write in sigma form

$sum=\sum _{i=1}^{\infty }\:42(\frac{3}{4})^{i-1}$

now, we can find sum

we can use formula

$sum=\frac{a}{1-r}$

now, we can plug values

we get

$sum=\frac{42}{1-\frac{3}{4}}$

$sum=168$

so, option-D.................Answer