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


the summation of 880 times one fourth to the i minus 1 power, from i equals 1 to infinity. ; the sum is divergent


the summation of 880 times one fourth to the i minus 1 power, from i equals 1 to infinity. ; the sum is 1,173


the summation of 880 times one fourth to the i power, from i equals 1 to infinity. ; the series is divergent


the summation of 880 times one fourth to the i power, from i equals 1 to infinity. ; the sum is 1,173

Answer 1

Answer: Second Option

"the summation of 880 times one fourth to the i minus 1 power, from i equals 1 to infinity. ; the sum is 1,173"

Step-by-step explanation:

We know that infinite geometrical series have the following form:

$\sum_{i=1}^{\infty}a_1(r)^{n-1}$

Where $a_1$ is the first term of the sequence and "r" is common ratio

In this case

$a_1 = 880\\\\r=\frac{1}{4}$

So the series is:

$\sum_{i=1}^{\infty}880(\frac{1}{4})^{n-1}$

By definition if we have a geometric series of the form

$\sum_{i=1}^{\infty}a_1(r)^{n-1}$

Then the series converges to  $\frac{a_1}{1-r}$   if $0<|r|<1$

In this case $r = \frac{1}{4}$ and $a_1=880$  then the series converges to $\frac{880}{1-\frac{1}{4}} = 1,173.3$

Finally the answer is the second option