Respuesta :
It is not possible to find the sum of an arithmetic sequence. Why not? because the terms keep getting bigger and bigger and bigger. For example: 1, 3, 5, 7, 9, 11, etc (add on 2 each time) is arithmetic. The sum of infinitely many terms of this sequence leads to infinity, which isn't a set number.
A geometric sequence on the other hand can have a finite sum. How? If we add on smaller and smaller geometric terms. Eg: 1, 1/4, 1/16, etc. Each term gets smaller and smaller so we slowly approach some fixed number as the infinite sum.
The formula to use for the infinite geometric series is
[tex]S = \frac{a}{1-r}[/tex]
where 'a' is the first term and r is the common ratio. This formula only works if
[tex] -1 < r < 1 [/tex]
In other words, r has to be between -1 and +1. The value of r cannot be equal to negative 1. The value of r cannot be equal to positive 1.
The sums of infinite arithmetic and geometric series are found by using the formula [tex]\rm S = \dfrac{a}{1-r}[/tex].
We have to find the sums of infinite arithmetic and geometric series.
According to the question,
The sum of an infinite series of a geometric progression is given by the general formula;
Where a is the first term of series, and r is the common ratio.
Let the first term of series be a and the common ratio of series as r.
Then,
The sum of an infinite series of a geometric progression is,
[tex]\rm S = \dfrac{a}{1-r}[/tex]
Where 'a' is the first term and r is the common ratio.
The value of r cannot be equal to positive 1.
Hence, the sums of infinite arithmetic and geometric series are found by using the formula [tex]\rm S = \dfrac{a}{1-r}[/tex].
To know more about Arithmetic Progression click the link given below.
https://brainly.com/question/24453114
