A geometric series 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 "common ratio". This series has been a topic of study for centuries due to its wide range of applications in various fields, including finance, physics, and engineering. In this section, we will delve deeper into the properties and characteristics of geometric series, exploring the formulae for the sum, nth term, and the behaviour of infinite series.
Introduction to Geometric Series
A geometric series is defined by its first term and the common ratio. The first term is often denoted as 'a', and the common ratio is represented by 'r'. The series is expressed as a sequence where each term is the product of the previous term and the common ratio.
For instance, consider the series: 5, 15, 45, 135, ... Here, the common ratio 'r' is 3, and the first term 'a' is 5.
Properties of Geometric Series
- Fixed Ratio: The most defining property of a geometric series is the constant ratio between consecutive terms. This ratio remains unchanged throughout the series.
- Convergence: An infinite geometric series converges (i.e., has a finite sum) if the absolute value of the common ratio 'r' is less than 1. If the common ratio is greater than or equal to 1, the series diverges.
- Multiplication by a Constant: If every term of a geometric series is multiplied by a constant, the resulting series is still geometric with the same common ratio.
- Sum of Finite Terms: The sum of the first 'n' terms of a geometric series can be found using the formula: Sn = a * (1 - rn) / (1 - r)
- Sum of Infinite Terms: The sum of an infinite geometric series, when it converges, is given by: S = a / (1 - r)
Formula for the nth Term of a Geometric Series
The nth term of a geometric series can be found using the formula: an = a * r(n - 1)
This formula allows us to determine any term in the series without having to list out all the preceding terms.
Example: To find the 7th term of a geometric series that starts with 2 and has a common ratio of 4: an = 2 * 4(7 - 1) = 2 * 4096 = 8192
Infinite Geometric Series
An infinite geometric series continues indefinitely. As mentioned, the sum of such a series converges if the absolute value of 'r' is less than 1. This is because as 'n' becomes very large, the term rn approaches zero, making the series sum finite.
Example: For an infinite series starting with 1 and having a common ratio of 0.5: S = 1 / (1 - 0.5) = 2
Practical Applications of Geometric Series
- Finance: Geometric series play a crucial role in calculating compound interest, annuities, and mortgages. The concept helps determine the future value of investments or the total amount payable.
- Physics: In scenarios like free-fall motion under gravity, the distances covered in equal time intervals form a geometric series.
- Engineering: Signal processing often uses geometric series to represent signals that decay or grow exponentially over time.
- Biology: Geometric series can model population growth, especially when a population multiplies by a constant factor in each time period.
Example Questions
1. Question: Find the sum of the first 6 terms of the geometric series that starts with 3 and has a common ratio of 2.
Solution: Using the formula: S6 = 3 * (1 - 26) / (1 - 2) S6 = 3 * (1 - 64) / (-1) S6 = 189
2. Question: Determine the sum of the infinite geometric series that starts with 4 and has a common ratio of 0.25.
Solution: Using the formula: S = 4 / (1 - 0.25) S = 5.33
FAQ
Yes, a geometric series can have a common ratio of zero. When this happens, every term after the first term will be zero. This is because each term is obtained by multiplying the previous term by the common ratio. If the common ratio is zero, then all subsequent terms will be zero, regardless of the value of the first term. For example, the series 5, 0, 0, 0,... has a first term of 5 and a common ratio of 0.
If every term of a geometric series is multiplied by a constant (k), the sum of the series will also be multiplied by that constant. This is because the sum is a combination of its terms. So, if each term is multiplied by k, the sum will increase by a factor of k. Mathematically, if S is the original sum, then the new sum after multiplying each term by k will be kS.
The formula for the sum of an infinite geometric series, S = a / (1 - r), is derived based on the assumption that the series converges to a finite value. For the series to converge, the terms must get progressively smaller, approaching zero. This happens only when the absolute value of the common ratio (r) is less than 1. If the absolute value of r is 1 or greater, the terms do not approach zero; instead, they either remain constant (if r is 1 or -1) or get larger (if the absolute value of r is greater than 1). In such cases, the series does not have a finite sum, making the formula inapplicable.
The common ratio (r) in a geometric series is determined by dividing any term by its preceding term. In other words, r = an / a(n-1), where an is the nth term and a(n-1) is the term before it. This ratio remains constant throughout the series. For instance, in the series 2, 6, 18, 54,... the common ratio is 6/2 = 3. It's crucial to ensure that this ratio remains consistent for all consecutive terms to confirm that the sequence is indeed geometric.
When the common ratio (r) of an infinite geometric series is greater than 1, the series does not converge to a finite value. Instead, it diverges, meaning the sum becomes infinitely large as more terms are added. This is because each successive term of the series is larger than the previous one, leading to an ever-increasing sum. In practical terms, this means that the sum of such a series is undefined or infinite. It's essential to recognise this property when working with geometric series, as it helps determine whether a series has a finite sum or not.
Practice Questions
To find the sum of the first 4 terms of the geometric series, we use the formula: Sn = a(1 - rn) / (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.
Substituting in the given values: S4 = 5(1 - 34) / (1 - 3) S4 = 5(1 - 81) / (-2) S4 = 5(-80) / (-2) S4 = 200
Therefore, the sum of the first 4 terms of the series is 200.
This is a geometric series problem where the first term a is £10,000 and the common ratio r is 1.10 (representing a 10% increase).
To find the total profit over 5 years, we need to find the sum of the first 5 terms using the formula: Sn = a(1 - rn) / (1 - r)
Substituting in the given values: S5 = 10,000(1 - 1.105) / (1 - 1.10) S5 = 10,000(1 - 1.61051) / (-0.10) S5 = 10,000(-0.61051) / (-0.10) S5 = £61,051
Therefore, the total profit over the 5 years will be £61,051.
