How can the sums of infinite arithmetic and geometric series be found?

The sums of infinite arithmetic and geometric series are found by using the formula .

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,

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 .

To know more about Arithmetic Progression click the link given below.

link

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

where 'a' is the first term and r is the common ratio. This formula only works if

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.

1. x> -5

2. x< 8

step-by-step explanation:

1. 2x> -10

divide by 2 from both sides of equation.

2x/2> -10/2

simplify, to find the answer.

-10/2=-5

x> -5 is the correct answer.

2. 3x< 24

divide by 3 from both sides of equation.

3x/3< 24/3

simplify, to find the answer.

24/3=8

x< 8 is the correct answer.

i hope this you, and have a wonderful day!