You can use the formula for evaluation of infinite geometric series.
The sum of the given infinite geometric series is 2500
What is a geometric sequence?
There are three parameters which differentiate between which geometric sequence we're taking about.
The first parameter is the initial value of the sequence.
The second parameter is the quantity by which we multiply previous term to get the next term.
The third parameter is the length of the sequence. It can be finite or infinite.
Suppose the initial term of a geometric sequence is [tex]a[/tex] and the term by which we multiply the previous term to get the next term is [tex]r[/tex]
Then the sequence would look like
[tex]a, ar, ar^2, ar^3...[/tex]
(till the terms to which it is defined)
What is a geometric series?
When all the terms of a geometric sequence are added, then that expression is called geometric series.
What is the sum of an infinite geometric series?
The sum of an infinite geometric series having the initial term "a" and the multiplication per term is done by "r" such that r < 1
is given by
[tex]S_{\infty} = \dfrac{a}{1-r}[/tex]
Using the above formula for finding the sum of the given geometric series
Since the given geometric series is 500 + 400 + 320 + ...
Thus, a = 500
Since r = ar/a = 2nd term divided by first term
Thus,
[tex]r = 400/500 = 0.8[/tex] < 1
thus, its sum is given by:
[tex]S_{\infty} = \dfrac{a}{1-r} = \dfrac{500}{1-0.8} = \dfrac{500}{0.2} = 2500[/tex]
Thus,
The sum of the given infinite geometric series is 2500
Learn more about sum of geometric series here;
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