Geometric Series Formula & Definitions

In mathematics, a geometric series is a series with a constant ratio between successive terms. For example, the sequence 2, 4, 8, 16, 32,… is a geometric series with common ratio 2. Geometric series are one of the simplest examples of infinite series with finite sums. In this blog post, we will explore the geometric series formula and some key definitions. We will also look at some examples of how to use the formula. By the end of this post, you should have a good understanding of what a geometric series is and how to use the formula.

What is a Geometric Series?

A geometric series is a mathematical series with a common ratio between successive terms. In a geometric series, each term after the first is equal to the previous term multiplied by a constant or common ratio. A geometric series converges if the absolute value of the common ratio is less than one and diverges if the common ratio is greater than one.

Geometric Series Formula

A geometric series is a mathematical series with a common ratio between successive terms. Geometric series are represented by the following formula:

S = a + ar + ar^2 + …

Where:

S is the sum of the geometric series
a is the first term of the geometric series
r is the common ratio between successive terms of the geometric series.

The sum of an infinite geometric series is given by the following formula:

S = a/(1-r)

Convergence of Geometric Series

To understand the Convergence of a Geometric Series, we must first understand what a series is. A series is the sum of the terms of a sequence. In other words, it is the mathematical way of adding numbers in succession. For example:

The above series converges because it approaches a finite limit as n goes to infinity. This limit is called the sum of the series and is represented by:

S = a1 + a2 + a3 + …+ an

In order for a series to converge, three conditions must be met:
(i) The sequence of partial sums must approach some finite limit as n goes to infinity.
(ii) This limit must be independent of the particular arrangement of terms in the original sequence.
(iii) The limit must be equal to the sum of the terms in the original sequence.
The first two conditions are fairly easy to grasp but let’s look at each one in turn with respect to our example above.
As n goes to infinity, each successive term in our example becomes smaller than the one before it. Therefore, the sequence of partial sums will approach some finite limit as n goes toward infinity (condition i).
Now let’s rearrange our terms to see if condition ii is satisfied:

What is a Geometric Sequence?

Geometric sequences are defined recursively. That is, the first term in the sequence is given, and each subsequent term is found by multiplying the previous one by a common ratio. For example, if the first term in a geometric sequence is 2 and the common ratio is 3, then the second term would be 6 (2×3), the third term would be 18 (6×3), and so on.

The general form of a geometric sequence is:

a, ar, ar^2, … , where a is the first term in the sequence and r is the common ratio

How to Find the Sum of a Geometric Series

To find the sum of a geometric series, you need to know the common ratio between terms and the number of terms in the series. The common ratio is the number that each term in the series is multiplied by to get the next term. For example, if the common ratio is 2, then the first term would be multiplied by 2 to get the second term, the second term would be multiplied by 2 to get the third term, and so on.

The sum of a geometric series is found using this formula:

S = a?(1 – r?)/(1 – r)

where S is the sum of the series, a? is the first term in the series, r is the common ratio, and n is the number of terms in the series.

For example, let’s say we have a geometric series with a first term of 2 (a? = 2), a common ratio of 3 (r = 3), and 4 terms in total (n = 4). We can plug those values into our formula to find that:

S = 2(1 – 3?)/(1 – 3)
= 2(1 – 81)/(-2)
= 2/-2 + 81/2
= 0 + 40.5
= 40.5

Examples of Geometric Sequences

A geometric sequence is a sequence in which each successive term is obtained by multiplying the previous term by a fixed non-zero number, called the common ratio.

For example, the sequence 2, 6, 18, 54,… is a geometric sequence with common ratio 3.

The first few terms of a geometric sequence can often be found by inspection. For example, the third term of the sequence above is 18 because it is 3 times 6 (i.e., 3×2=6 and 3×6=18).

To find the nth term of a geometric sequence with initial value a1 and common ratio r, we use the formula:

an = arn?1

where n is the position of the desired term in the sequence (first term being n = 1). Therefore, to find the 100th term of our example sequence above, we would plug in n = 100 to get:

a100 = 3 × 99 = 297

The sum of all terms in a geometric series with initial value a1 and common ratio r is given by:

S_n=frac{a_1(1-r^n)}{1-r} quad textrm{if } |r| < 1

Conclusion

In mathematics, a geometric series is a series with a constant ratio between successive terms. Geometric series are one of the simplest examples of infinite series with finite sums. Interestingly, they turn up in many different settings, from compound interest to fractals. In this article, we looked at the formula for the sum of a geometric series and some of its key properties. We also saw how it can be used to solve problems involving growth and decay.