Geometric Series Formula
Geometric Series Formula
Number sequences follow certain patterns and laws. Any routine pattern that is transferable from one term to the next may be this pattern. The Geometric Series Formula is one in which more multiplication is required to obtain the following term. The sum of the sequence’s terms can be obtained by adding or subtracting symbols. The geometric series will result from this. One such geometric series is 1 + 2 + 4 + 8 + 16 for instance.
The geometric progression is another name for a geometric series. It is a series where the second term is obtained by multiplying the first term by a predetermined amount. This operation is repeated until the series has the necessary number of terms. Such a rise occurs in a particular way, resulting in geometric development.
The geometric series formula will determine the general term and the total of all its terms. For instance, if students add 2 to the first number in the series above, we will receive the second number, and so on.
Such Geometric Series Formula follow the straightforward rule that you must multiply one term by a certain integer to reach the next. As a result, students can produce any number of terms for this series. Thus, students can learn about the usual ratios or fixed values for multiplication by looking at such a series. It is represented by the letter “r.” One can study the Geometric Series Formula on the Extramarks website or mobile application.
What is a Geometric Series?
The total of a geometric sequence’s finite or infinite terms is known as a geometric series. The analogous geometric series is a + ar + ar2 +…, arn-1 + for the geometric sequence a, ar, ar2,…, arn-1,… The word “series” is known to mean “sum.”
The Geometric Series Formula specifically refers to the total of phrases with a common ratio between every pair of neighbouring terms. Geometric series can be of two different types: finite and infinite. Below are a few illustrations of the Geometric Series Formula.
In a finite Geometric Series Formula, 1/2 + 1/4 +…. + 1/8192, the first term, a, is equal to 1/2 and the common ratio, r, is equal to 1/2. In an infinite geometric series, -4 + 2 – 1 + 1/2 – 1/4 +…, the first term, a, is equal to -4 and the common ratio, r, is equal to -1/2.
Geometric Series Formula
Let’s review what a Geometric Series Formula is first before learning the formula. Every two successive terms in the series (sum of terms) have the same ratio. The formula for the geometric series includes A formula to determine a Geometric Series Formula nth term, calculate a finite geometric series’ sum, and calculate the total of an endless geometric sequence.
Geometric Series Formulas
Calculating the Geometric Series Formula is necessary to understand the compounding that takes place over time. By using the geometric progression’s mean, it illustrates the fundamental behaviour of the progression. For instance, it is simple to use the geometric mean to examine the growth of bacteria. In other words, the compounding becomes more critical the longer the time horizon or the more different the values in the series, and as a result, the geometric mean is a better choice.
The formulas to determine the nth term, the sum of n terms, and the sum of infinite terms are all included in the formulas for geometric series. Consider a Geometric Series Formula where the common ratio is r, and the first term is a.
a + ar + ar2 + ar3 + …
Convergence of Geometric Series
Any finite geometric series converges eventually. However, the common ratio’s value determines whether or not an infinite geometric series will eventually converge. A, Ar, Ar2,… is a part of an infinite geometric series.
converges when |r| 1. Thus, one can use the formula a / to find its sum (1 – r).
diverges when |r| > 1. Therefore, it is not possible to determine its total in this situation.
Examples Using Formula for a Geometric Series
Example 1: Find the 10th term of the geometric series 1 + 4 + 16 + 64 + …
To find The 10th term of the given geometric series.
In the given series,
The first term, a = 1.
The common ratio r = 4 / 1 (or) 16 / 4 (or) 64 / 16 = 4.
Using the Geometric Series Formula, the nth term is found using:
nth term = an rn-1
Substitute n = 10, a = 1, and r = 4 in the above formula:
10th term = 1 × 410-1 = 49 = 262,144
Answer: The 10th term of the given geometric sequence = 262,144.
Example 2: Find the sum of the following geometric series: i) 1 + (1/3) + (1/9) + … + (1/2187)
ii) 1 + (1/3) + (1/9) + … using the Geometric Series Formula:
To find The sum of the given two geometric series.
In both of the given series,
the first term, a = 1.
The common ratio r = 1 / 3.
- i) In the given series,
nth term = 1 / 2187
a rn-1 = 1 / 37
1 × (1 / 3) n-1 = 3-7
3-n + 1 = 3-7
-n + 1 = -7
-n = -8
n = 8
So we need to find the sum of the first 8 terms of the given series.
Using the sum of the finite geometric series formula:
Sum of n terms = a (1 – rn) / (1 – r)
Sum of 8 terms = 1 (1 – (1/3)8) / (1 – 1/3)
= (1 – (1 / 6561)) / (2 / 3)
= (6560 / 6561) × (3 / 2)
= 3280 / 2187
ii) The given series is an infinite Geometric Series Formula.
Using the sum of the infinite Geometric Series Formula:
Sum of infinite geometric series = a / (1 – r)
Sum of the given infinite geometric series
= 1 / (1 – (1/3))
= 1 / (2 / 3)
= 3 / 2
Answer: i) Sum = 3280 / 2187 and ii) Sum = 3 / 2
Example 3: Calculate the sum of the finite geometric series if a = 5, r = 1.5 and n = 10.
To find: the sum of the Geometric Series Formula
Given: a = 5, r = 1.5, n = 10
sn = a(1−rn)/(1−r)
The sum of ten terms is given by S10 = 5(1−(1.5)10)/(1−1.5)
Answer: The sum of the geometric series is 566.65.
FAQs (Frequently Asked Questions)
1. What Are the Mathematical Geometric Series Formulas?
The formulas for geometric series can be used to determine the sum of a finite geometric sequence, the sum of an infinite geometric series, and the nth term of a geometric sequence. The initial term “a” and the common ratio “r” in these Geometric Series Formula are given as
A rn-1 = nth term
The Sum of terms is equal to a (1 – rn) (1 – r)
The Sum of all geometric series is equal to a. (1 – r)
2. Is it always possible to determine the sum of a geometric series?
Students can always find the sum of a geometric series if it is finite. However, only when its common ratio’s absolute value is less than one can its total be calculated.
3. What is the point at which a geometric sequence converges?
The letter “r” represents the common ratio between each pair of consecutive entries in a geometric series. Only when |r| one does an infinite geometric series converge, and only then can its sum be determined.
4. What Purposes Do Geometric Series Serve?
Formulas for geometric series are often used in mathematics. These have significant applications in finance, queueing theory, computer science, biology, economics, and engineering.
5. How does the"r" work in Geometric Series Formula?
The term “r” in the formula for geometric series stands for the common ratio. The following are the formulas for geometric series with “n” terms with the term “a” as the first term: The equation for the nth term: A rn-1 = nth term, the Sum of terms is equal to a (1 – rn) (1 – r) and the Sum of all geometric series is equal to a. (1 – r)
6. What distinguishes geometric sequence from geometric series?
A geometric sequence is a collection of terms in which the ratio of every two successive terms is the same, whereas a geometric series is the geometric sequence’s “sum.”