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Discover the world of Geometric Sequences: Unravel the patterns and properties of these mathematical progressions with our comprehensive guide and learn how these sequences differ from arithmetic progressions and their significance in mathematics.
Contents
What is a Geometric Sequence?
A geometric sequence, also known as a geometric progression, is a sequence of numbers in which each term is found by multiplying the previous term by a fixed, non-zero number called the “common ratio.” This common ratio is denoted by the letter “r.”
The general form of a geometric sequence is:
a, ar, ar^2, ar^3, ar^4, …
Where:
- “a” is the first term in the sequence.
- “r” is the common ratio, which is a constant value.
Each term in a geometric sequence is obtained by multiplying the previous term by the common ratio. For example, to find the third term, you would multiply the second term by “r,” and to find the fourth term, you would multiply the third term by “r,” and so on.
Geometric sequences can be either finite or infinite. A finite geometric sequence has a specific number of terms, while an infinite geometric sequence continues indefinitely. The terms of an infinite geometric sequence become increasingly larger or smaller, depending on whether the common ratio is greater than 1 (leading to growth) or between 0 and 1 (leading to decay).
The formula for finding the nth term (an) of a geometric sequence is:
an = a * r^(n-1)
Where:
- “an” is the nth term of the sequence.
- “a” is the first term.
- “r” is the common ratio.
- “n” is the position of the term in the sequence.
Geometric sequences have various applications in mathematics, science, and finance, such as modeling exponential growth or decay, calculating compound interest, and understanding population dynamics.
Geometric Sequence Formulas
A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the “common ratio.” The general form of a geometric sequence is:
a, ar, ar^2, ar^3, …
Where:
- “a” is the first term in the sequence.
- “r” is the common ratio.
Here are some important formulas and concepts related to geometric sequences:
- nth Term Formula: The nth term of a geometric sequence can be found using the formula:
an = a * r^(n-1)
Where:
- “an” is the nth term.
- “a” is the first term.
- “r” is the common ratio.
- “n” is the position of the term in the sequence.
- Sum of the First n Terms (Partial Sum): The sum of the first n terms of a geometric sequence is given by the formula for the partial sum:
Sn = (a * (1 – r^n)) / (1 – r)
Where:
- “Sn” is the sum of the first n terms.
- “a” is the first term.
- “r” is the common ratio.
- “n” is the number of terms to be summed.
- Infinite Geometric Series: If the common ratio “r” is between -1 and 1 (i.e., -1 < r < 1), then the sum of an infinite geometric series can be calculated using the formula:
S∞ = a / (1 – r)
Where:
- “S∞” represents the sum of an infinite geometric series.
- Common Ratio (r) Determination: To find the common ratio of a geometric sequence, you can divide any term by the previous term. The common ratio is the result of this division. For example:
r = a2 / a1 = a3 / a2 = a4 / a3
These formulas and concepts are fundamental for working with geometric sequences and are commonly used in various mathematical and scientific applications.
Geometric Sequence vs Arithmetic Sequence
Here’s a tabular comparison between a geometric sequence and an arithmetic sequence:
on using regular alphabet characters:
| Property | Geometric Sequence | Arithmetic Sequence |
|---|---|---|
| General Form | a, ar, ar^2, ar^3, … | a, a + d, a + 2d, a + 3d, … |
| Common Ratio (r) | Multiplying by a constant ‘r’ | Adding a constant ‘d’ |
| Formula for ‘n’th Term | a_n = a * r^(n-1) | a_n = a + (n-1) * d |
| Sum of First ‘n’ Terms | S_n = a * (1 – r^n) / (1 – r) | S_n = (n/2) * [2a + (n-1)d] |
| Characteristic Behavior | Each term is obtained by multiplying the previous term by ‘r’. | Each term is obtained by adding a constant ‘d’ to the previous term. |
| Examples | 2, 6, 18, 54, … (with ‘a’ = 2 and ‘r’ = 3 as an example) | 3, 7, 11, 15, … (with ‘a’ = 3 and ‘d’ = 4 as an example) |
In a geometric sequence, each term is obtained by multiplying the previous term by a constant factor (common ratio), denoted as ‘r.’ This results in an exponential growth or decay pattern.
In an arithmetic sequence, each term is obtained by adding a constant value (common difference), denoted as ‘d,’ to the previous term. This results in a linear pattern with a constant rate of change.
Both types of sequences are used in mathematics and various applications, such as finance, physics, and computer science, to model different types of growth, progression, or patterns.
n^th Term of Geometric Sequence Formula
The nth term of a geometric sequence can be found using the following formula:
an = a1 * r^(n-1)
Where:
- “an” represents the nth term of the sequence.
- “a1” is the first term of the sequence.
- “r” is the common ratio between consecutive terms.
- “n” is the term number you want to find.
Sum of Finite Geometric Sequence Formula
The sum of a finite geometric sequence can be calculated using the following formula:
Sn = a1(1 – rn) / (1 – r)
Where:
- Sn is the sum of the first n terms of the geometric sequence.
- a1 is the first term of the sequence.
- r is the common ratio between the terms of the sequence.
- n is the number of terms you want to sum.
The formula allows you to find the sum of a geometric sequence without having to manually calculate each term and add them up individually.
Sum of Infinite Geometric Sequence Formula
The sum of an infinite geometric sequence, often denoted as “S” or “∑,” can be calculated using the following formula:
S = a / (1 – r)
Where:
S is the sum of the infinite geometric sequence.a is the first term of the sequence.r is the common ratio between the terms. (|r| < 1 for convergence)It’s important to note that this formula is valid only when the common ratio “r” has an absolute value less than 1, which is a condition for the series to converge. If the common ratio is greater than or equal to 1 in absolute value, the series does not have a finite sum.
Here’s an example to illustrate how to use this formula:
Suppose you have an infinite geometric sequence with the first term (a) equal to 2 and a common ratio (r) of 0.5. To find the sum of this infinite sequence, you can use the formula:
S = 2 / (1 – 0.5)
S = 2 / 0.5
S = 4
So, the sum of the infinite geometric sequence with a first term of 2 and a common ratio of 0.5 is 4.
Some Solved Examples of Geometric Sequence
Here are some of the examples of the geometric sequence.
Example 1: Find the 7th term of a geometric sequence with a common ratio of 2 and a first term of 3.
Given: First term (a) = 3 Common ratio (r) = 2
To find the 7th term (n = 7), you can use the formula for the nth term of a geometric sequence:
an = a * r^(n-1)
a7 = 3 * 2^(7-1)
a7 = 3 * 2^6
a7 = 3 * 64
a7 = 192
So, the 7th term of this geometric sequence is 192.
Example 2: Determine the sum of the first 5 terms of a geometric sequence with a first term of 2 and a common ratio of 3.
Given: First term (a) = 2 Common ratio (r) = 3 Number of terms to sum (n) = 5
To find the sum of the first 5 terms, you can use the formula for the sum of a geometric series:
Sn = a * (r^n – 1) / (r – 1)
S5 = 2 * (3^5 – 1) / (3 – 1)
S5 = 2 * (243 – 1) / 2
S5 = 2 * 242 / 2
S5 = 242
So, the sum of the first 5 terms of this geometric sequence is 242.
Example 3: Find the 6th term of a geometric sequence where the first term is 5/2 and the common ratio is 1/4.
Given: First term (a) = 5/2 Common ratio (r) = 1/4
To find the 6th term (n = 6), you can use the formula for the nth term of a geometric sequence:
an = a * r^(n-1)
a6 = (5/2) * (1/4)^(6-1)
a6 = (5/2) * (1/4)^5
a6 = (5/2) * (1/1024)
a6 = 5/2048
So, the 6th term of this geometric sequence is 5/2048.
These are the examples of geometric sequences with calculations using normal alphabetical characters.
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