Question

The population of a local species of flies can be found using an infinite geometric series where a1 = 940 and the common ratio is one fifth. Write the sum in sigma notation, and calculate the sum (if possible) that will be the upper limit of this population.

Answer 1

A geometric series is the sum of the terms of a geometric sequence of the form
`a, ar, ar^(2), ar^(3),...`,
where r is the common ratio, and a≠0 is the first term.

That is, the series is
`a+ar+ar^(2)+ar^(3),...`

In sigma notation, the series is written as:

∞
∑
`a r^(k)`
k=0
--------------------------------------------------------------------------------------------------

The geometric series of the form
∞
∑
`a r^(k)`, converges to
`(a)/(1-r)` if |r|<1
k=0

and diverges otherwise.
--------------------------------------------------------------------------------------------------

in our problem, a , the first term is equal to 940, and the common ratio is |1/5|<1,

thus the series converges to:

`(a)/(1-r)=(940)/(1-1/5)=(940)/(4/5)=752`


Answer:

∞
∑
`940 (1/5)^(k)`=752 ( the upper limit of the population is 752)
k=0