In mathematics, the common ratio refers to the constant value by which each term in a geometric sequence is multiplied to obtain the next term. It is a fundamental concept in the study of sequences and series, particularly in geometric progressions.
The concept of the common ratio can be traced back to ancient civilizations such as Babylon and Egypt, where mathematicians observed patterns in numbers and geometric shapes. However, the formal study of geometric progressions and the common ratio began in ancient Greece with the works of mathematicians like Euclid and Pythagoras.
The concept of the common ratio is typically introduced in middle school or early high school mathematics, depending on the curriculum. It is an important topic in algebra and is often covered in courses such as Algebra 1 or Algebra 2.
The common ratio contains several key knowledge points, which are explained step by step:
Geometric Sequence: A geometric sequence is a sequence of numbers in which each term is obtained by multiplying the previous term by a fixed, non-zero value called the common ratio.
Formula for the nth Term: The nth term of a geometric sequence can be found using the formula: an = a1 * r^(n-1), where an represents the nth term, a1 is the first term, r is the common ratio, and n is the position of the term in the sequence.
Sum of a Geometric Series: The sum of a finite geometric series can be calculated using the formula: Sn = a1 * (1 - r^n) / (1 - r), where Sn represents the sum of the first n terms, a1 is the first term, r is the common ratio, and n is the number of terms.
There are two types of common ratio:
Positive Common Ratio: When the common ratio is greater than 1, each term in the sequence becomes larger than the previous term.
Negative Common Ratio: When the common ratio is between -1 and 0, each term in the sequence becomes smaller than the previous term.
The common ratio possesses several properties:
Determines the Growth or Decay: The common ratio determines whether the sequence grows or decays. If the common ratio is greater than 1, the sequence grows exponentially. If the common ratio is between -1 and 0, the sequence decays exponentially.
Relationship between Terms: Each term in a geometric sequence is related to the previous term by multiplication with the common ratio.
Uniqueness: A geometric sequence is uniquely determined by its first term and the common ratio.
To find or calculate the common ratio, you need at least two consecutive terms of a geometric sequence. The common ratio can be obtained by dividing any term by its previous term.
The formula for the common ratio (r) in a geometric sequence is given by:
r = a2 / a1
where a1 is the first term and a2 is the second term of the sequence.
To apply the common ratio formula, substitute the values of the first term (a1) and the second term (a2) into the formula:
r = a2 / a1
By calculating this ratio, you can determine the common ratio of the geometric sequence.
The symbol commonly used to represent the common ratio is 'r'.
There are several methods for finding the common ratio:
Division Method: Divide any term of the sequence by its previous term to obtain the common ratio.
Formula Method: Use the formula r = a2 / a1, where a1 is the first term and a2 is the second term of the sequence.
Pattern Method: Observe the pattern in the sequence and identify the constant value by which each term is multiplied to obtain the next term.
Example 1: Find the common ratio of the geometric sequence: 2, 4, 8, 16, ...
Solution: Divide any term by its previous term: 4 / 2 = 2. Therefore, the common ratio is 2.
Example 2: Given a geometric sequence with a common ratio of -0.5 and a first term of 16, find the fifth term.
Solution: Use the formula for the nth term: an = a1 * r^(n-1) a5 = 16 * (-0.5)^(5-1) = 16 * (-0.5)^4 = 16 * 0.0625 = 1
Therefore, the fifth term of the sequence is 1.
Example 3: Find the sum of the geometric series: 3, 6, 12, 24, ... up to the 5th term.
Solution: Use the formula for the sum of a geometric series: Sn = a1 * (1 - r^n) / (1 - r) Sn = 3 * (1 - 2^5) / (1 - 2) = 3 * (1 - 32) / (1 - 2) = 3 * (-31) / (-1) = 93
Therefore, the sum of the first 5 terms of the series is 93.
Find the common ratio of the geometric sequence: 1, 3, 9, 27, ...
Given a geometric sequence with a common ratio of 0.2 and a first term of 100, find the tenth term.
Find the sum of the geometric series: 2, 4, 8, 16, ... up to the 6th term.
Question: What is the common ratio? Answer: The common ratio is the constant value by which each term in a geometric sequence is multiplied to obtain the next term.