Definition of a Geometric Sequence
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the ratio. Mathematically, a geometric sequence can be expressed as:
a, ar, ar2, ar3, ar4, ...
- a: The first term in the sequence.
- r: The common ratio, which is a constant value multiplied to a term to get the subsequent term.
Characteristics of Geometric Sequences
- Fixed Multiplicative Pattern: The terms of the sequence exhibit a consistent multiplicative pattern, wherein each term is a product of the preceding term and the common ratio.
- Infinite or Finite: Geometric sequences can be infinite or finite, depending on the context or problem at hand.
- Direction: The sequence can be increasing, decreasing, or oscillating, contingent upon the value of the common ratio.
Example 1: Identifying a Geometric Sequence
Consider the sequence: 3, 6, 12, 24, 48, ...
Here, each term is obtained by multiplying the previous term by 2. Thus, it is a geometric sequence with a common ratio of 2.
Common Ratio: The Heart of a Geometric Sequence
The common ratio is a pivotal element in a geometric sequence, influencing its behaviour and properties. It is denoted by r and is the factor by which we multiply one term to get to the next.
Determining the Common Ratio
To find the common ratio in a geometric sequence, divide any term by its preceding term. Mathematically,
r = an / a(n-1)
where:
- an is the nth term.
- a(n-1) is the term before the nth term.
Implications of the Common Ratio
- r > 1: The sequence increases exponentially.
- 0 < r < 1: The sequence decreases exponentially.
- r < 0: The sequence oscillates between positive and negative terms.
Example 2: Determining the Common Ratio
Given the sequence: 5, 10, 20, 40, ...
To find the common ratio, divide the second term by the first term:
r = 10 / 5 = 2
Hence, the common ratio for this sequence is 2.
Exploring Geometric Sequences Through Examples
Example 3: Analysing a Geometric Sequence
Consider the sequence: 4, 12, 36, 108, ...
- First Term (a): 4
- Common Ratio (r): 3 (since 12/4 = 3)
Let’s find the 5th term of the sequence.
To find the nth term of a geometric sequence, we use the formula:
an = a * r(n-1)
Substituting the values we have:
a5 = 4 * 3(5-1) a5 = 4 * 34 a5 = 4 * 81 a5 = 324
Thus, the 5th term of the sequence is 324.
Example 4: Application in Real-World Scenario
Imagine you invested £1000 in a scheme where the investment doubles every year. This forms a geometric sequence: £1000, £2000, £4000, £8000, ...
- First Term (a): £1000
- Common Ratio (r): 2
If you wanted to calculate your investment after 6 years, you would substitute these values into the formula:
a6 = £1000 * 2(6-1) a6 = £1000 * 25 a6 = £1000 * 32 a6 = £32000
After 6 years, your investment would grow to £32000.
Reflections on Geometric Sequences
Geometric sequences permeate various aspects of mathematics and its applications, especially in financial maths, physics, and engineering. The understanding of the basic definition and the pivotal role of the common ratio provides a foundation to explore more advanced concepts and applications, which will be further explored in subsequent topics.
FAQ
Geometric sequences are widely used in various real-world applications, especially in financial mathematics, physics, and computer science. In finance, geometric sequences can model investment growth, loan repayments, and depreciation. For instance, compound interest is calculated using a formula derived from a geometric sequence. In physics, geometric sequences can describe exponential decay, such as radioactive decay, or growth, such as population growth in a model with constant reproduction rate. In computer science, algorithms, particularly those related to search and sort, often have efficiencies that can be expressed using geometric sequences.
The sum of an infinite geometric sequence, also known as an infinite geometric series, is defined only when the common ratio "r" is between -1 and 1 (i.e., -1 < r < 1). The formula to find the sum (S) of an infinite geometric sequence is S = a / (1 - r), where "a" is the first term. If the common ratio is outside of this range, the sum does not converge to a finite value and therefore is not defined. The rationale behind this is that when -1 < r < 1, the terms of the sequence get progressively smaller, approaching zero, allowing for a finite sum.
Absolutely, a geometric sequence can have fractional or decimal terms. The common ratio "r" can be a fraction or a decimal, and the first term "a" can also be fractional or decimal. For instance, if a = 1/2 and r = 0.5, the sequence will be 1/2, 1/4, 1/8, 1/16, ... and so on. Similarly, if a and r are fractions, such as a = 3/4 and r = 2/3, the sequence will be 3/4, 1/2, 1/3, 2/9, ... and so on. The rules and formulas for geometric sequences apply in the same way, regardless of whether the terms and ratio are whole numbers, fractions, or decimals.
Yes, a geometric sequence can have a common ratio of 1 or 0, but these cases are somewhat trivial. If the common ratio, r, is 1, every term in the sequence will be the same as the first term, creating a constant sequence. For example, if a = 5 and r = 1, the sequence will be 5, 5, 5, 5, ... and so on. If r is 0, every term after the first term will be zero, regardless of the value of the first term. For instance, if a = 5 and r = 0, the sequence will be 5, 0, 0, 0, ... and so on.
The common ratio, denoted as "r", significantly influences the behaviour of a geometric sequence. If r > 1, the sequence will increase exponentially, meaning each subsequent term will be larger than the previous one, leading to a rapidly increasing sequence. If 0 < r < 1, the sequence will decrease exponentially, with each term becoming smaller, yet never reaching zero. If r < 0, the sequence will oscillate between positive and negative values, creating an alternating sequence. The absolute value of r will determine the rate of increase or decrease, with larger absolute values leading to steeper inclines or declines in the sequence.
Practice Questions
The 6th term of a geometric sequence can be found using the formula a_n = a * r(n-1), where "a" is the first term, "r" is the common ratio, and "n" is the term number. Substituting the given values into the formula, we get a_6 = 3 * 2(6-1) = 3 * 25 = 3 * 32 = 96. Therefore, the 6th term of the geometric sequence is 96.
To find the value of k, we can use the property of geometric sequences that states the square of the middle term is equal to the product of its two adjacent terms. Mathematically, this is expressed as (term2)2 = term1 * term3. Substituting the given values into this formula, we get k2 = 5 * 45 = 225. Taking the square root on both sides, we find that k = 15, since we are considering the positive root in the context of sequences. Therefore, the value of k is 15.
