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Sum of a geometric sequence
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Sum of a Geometric Sequence
Introduction
A geometric sequence is a fundamental concept in mathematics, particularly within the IB Mathematics: Analysis and Approaches Standard Level (AA SL) curriculum. Understanding the sum of a geometric sequence is essential for solving various real-world problems, including financial calculations and population growth models. This article delves into the intricacies of geometric sequences, providing a comprehensive overview tailored to IB students.
Key Concepts
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 constant called the common ratio ($r$). The general form of a geometric sequence is:
$$a, ar, ar^2, ar^3, \dots$$
where:
- $a$ is the first term.
- $r$ is the common ratio.
Formula for the Sum of a Finite Geometric Sequence
The sum ($S_n$) of the first $n$ terms of a geometric sequence is given by:
$$S_n = a \frac{1 - r^n}{1 - r}, \quad \text{for } r \neq 1$$
Where:
- $a$ is the first term.
- $r$ is the common ratio.
- $n$ is the number of terms.
Derivation of the Sum Formula
Consider the sum of the first $n$ terms:
$$S_n = a + ar + ar^2 + \dots + ar^{n-1}$$
Multiply both sides by $r$:
$$rS_n = ar + ar^2 + \dots + ar^{n}$$
Subtract the second equation from the first:
$$S_n - rS_n = a - ar^n$$
Factor out $S_n$:
$$(1 - r)S_n = a(1 - r^n)$$
Solve for $S_n$:
$$S_n = a \frac{1 - r^n}{1 - r}$$
Infinite Geometric Series
When the number of terms approaches infinity ($n \to \infty$), and if $|r| < 1$, the sum of the infinite geometric series converges to:
$$S_{\infty} = \frac{a}{1 - r}$$
This is particularly useful in applications where the sequence continues indefinitely, and the common ratio diminishes the contribution of each subsequent term.
Applications of Geometric Sequences and Their Sums
Geometric sequences and their sums are prevalent in various fields:
- Finance: Calculating compound interest involves geometric series.
- Population Growth: Modeling populations with constant growth rates.
- Physics: Analyzing waves and oscillations where amplitude decreases geometrically.
Examples and Problem Solving
Example 1: Find the sum of the first 5 terms of the geometric sequence where $a = 3$ and $r = 2$.
$$S_5 = 3 \frac{1 - 2^5}{1 - 2} = 3 \frac{1 - 32}{-1} = 3 \times 31 = 93$$
Example 2: Determine the sum of an infinite geometric series with $a = 100$ and $r = \frac{1}{2}$.
$$S_{\infty} = \frac{100}{1 - \frac{1}{2}} = \frac{100}{\frac{1}{2}} = 200$$
Common Mistakes and How to Avoid Them
- Incorrect Identification of Common Ratio: Ensure that $r$ is consistent between consecutive terms.
- Applying the Sum Formula When $r = 1$: The formula is undefined for $r = 1$; instead, $S_n = na$.
- Misapplying the Infinite Sum Formula: Only applicable when $|r| < 1$.
Practice Problems
- Find the sum of the first 8 terms of a geometric sequence where the first term is 5 and the common ratio is 3.
- If the sum of an infinite geometric series is 50 and the first term is 20, find the common ratio.
- Calculate the sum of the first 10 terms of a geometric sequence with $a = 2$ and $r = -1.5$.
Comparison Table
| Aspect | Finite Geometric Series | Infinite Geometric Series |
| Number of Terms | Limited to $n$ terms | Extends to infinity |
| Sum Formula | $S_n = a \frac{1 - r^n}{1 - r}$ | $S_{\infty} = \frac{a}{1 - r}$ |
| Conditions for Convergence | N/A | $|r| < 1$ |
| Applications | Finite financial investments, loan repayments | Perpetual bonds, infinite resource models |
Summary and Key Takeaways
- A geometric sequence multiplies by a constant ratio between terms.
- The sum of a finite geometric sequence is calculated using $S_n = a \frac{1 - r^n}{1 - r}$.
- An infinite geometric series converges to $S_{\infty} = \frac{a}{1 - r}$ only if $|r| < 1$.
- Applications span finance, population studies, and physics.
- Understanding common ratios and convergence conditions is crucial for accurate problem-solving.
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Examiner Tip
Tips
Use the mnemonic "S.A.F.E." to remember the Sum formula: S = $a \frac{1 - r^n}{1 - r}$. Additionally, sketching the first few terms can help visualize whether the series converges or diverges. Practicing with different values of $r$ reinforces understanding of how the common ratio affects the sum.
Did You Know
Did You Know
Geometric sequences are not only pivotal in mathematics but also play a crucial role in computer science algorithms, such as those used in searching and sorting. Additionally, the concept of geometric series is fundamental in calculus, particularly in the study of power series and Taylor expansions, which are essential for approximating complex functions.
Common Mistakes
Common Mistakes
Misidentifying the Common Ratio: Students often confuse addition with multiplication when determining $r$. For example, in the sequence 2, 6, 18, the common ratio is 3, not 4 (2 + 4 = 6, but 6 × 3 = 18). Correct approach: Divide a term by its preceding term to find $r$.
Ignoring the Convergence Condition: Applying the infinite sum formula without checking if $|r| < 1$ leads to incorrect results. Always verify the common ratio before using $S_{\infty} = \frac{a}{1 - r}$.
FAQ
What is a geometric sequence?
A geometric sequence is a sequence of numbers where each term after the first is obtained by multiplying the previous term by a constant called the common ratio ($r$).
How do you find the sum of the first n terms of a geometric sequence?
Use the formula $S_n = a \frac{1 - r^n}{1 - r}$, where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.
When does an infinite geometric series converge?
An infinite geometric series converges if the absolute value of the common ratio is less than 1 ($|r| < 1$).
What is the sum of an infinite geometric series?
If $|r| < 1$, the sum of an infinite geometric series is $S_{\infty} = \frac{a}{1 - r}$.
Can the common ratio be negative?
Yes, the common ratio can be negative, which causes the terms to alternate in sign.
What happens when $r = 1$ in a geometric sequence?
When $r = 1$, all terms in the sequence are equal to the first term, and the sum of the first $n$ terms is $S_n = na$.
1.
Calculus
2.
Number and Algebra
3.
Functions
4.
Geometry and Trigonometry
5.
Exploration and Mathematical Investigations
6.
Statistics and Probability
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