Welcome to Sequences and Series!
Hello! This chapter is all about spotting and describing patterns. Don't worry if numbers seem tricky; mathematics is full of beautiful, predictable structures, and "Sequences and Series" gives you the tools to understand and predict them.
We will learn how numbers in a list (a sequence) behave, and how to quickly add them all up (a series), even if the list is incredibly long. These concepts are fundamental for understanding everything from compound interest to predicting populations!
Why Study Sequences and Series?
- Finance: Calculating loan repayments or savings growth relies on geometric sequences.
- Computer Science: Algorithms often use patterns defined by sequences.
- Modeling: Predicting the decay of radioactive material or the spread of a virus uses these ideas.
Section 1: The Foundations – Sequences and Sigma Notation
What is a Sequence?
A Sequence is simply an ordered list of numbers. Each number in the sequence is called a term. We usually denote the \(n\)-th term (the term in the \(n\)-th position) as \(u_n\).
Example: 2, 4, 6, 8, 10, ...
Here, the first term \(u_1 = 2\), the second term \(u_2 = 4\), and so on.
Defining Sequences: Two Main Types of Rules
1. Term-to-Term Rule (Recurrence Relation)
This rule tells you how to get the next term from the previous term. You must always be given the starting term (\(u_1\)).
Example: If \(u_{n+1} = u_n + 3\) and \(u_1 = 5\).
\(u_2 = u_1 + 3 = 5 + 3 = 8\)
\(u_3 = u_2 + 3 = 8 + 3 = 11\)
The sequence is 5, 8, 11, 14, ...
2. Position-to-Term Rule (The \(n\)-th Term Formula)
This rule lets you calculate any term based on its position \(n\), without needing to know the previous terms. This is usually the most useful rule!
Example: If the rule is \(u_n = 2n + 1\).
To find the 10th term (\(n=10\)): \(u_{10} = 2(10) + 1 = 21\).
Understanding Series and Sigma Notation (\(\sum\))
A Series is the sum of the terms in a sequence.
We use the Greek letter Sigma, \(\sum\), as a quick shorthand way of saying "sum everything up."
How to Read Sigma Notation
The notation looks like this: \[ \sum_{n=1}^{k} u_n \]
Breakdown:
- \(\sum\): Means "Sum."
- \(u_n\): This is the formula for the terms you are adding.
- \(n=1\): This is the starting value (the first term you calculate).
- \(k\): This is the ending value (the last term you calculate).
Example: Calculate \( \sum_{n=1}^{4} (3n) \)
This means:
(Calculate term for \(n=1\)) + (Calculate term for \(n=2\)) + (Calculate term for \(n=3\)) + (Calculate term for \(n=4\))
\(= (3(1)) + (3(2)) + (3(3)) + (3(4))\)
\(= 3 + 6 + 9 + 12 = 30\)
Quick Review: Sequences are lists; Series are sums. Sigma notation is the shorthand for summing up a series.
Section 2: Arithmetic Progressions (AP)
Don't worry about the formal name! An Arithmetic Progression (AP), or Arithmetic Sequence, is simply a sequence where the difference between consecutive terms is constant. We call this constant difference the Common Difference, denoted by \(d\).
Analogy: Think of climbing a very stable staircase. Each step (term) is exactly the same height (\(d\)) higher than the last.
1. The \(n\)-th Term of an AP
Let the first term be \(a\) (which is \(u_1\)).
- \(u_1 = a\)
- \(u_2 = a + d\)
- \(u_3 = a + 2d\)
- \(u_4 = a + 3d\)
Notice the pattern: the number of times you add \(d\) is always one less than the term number \(n\).
The formula for the \(n\)-th term of an AP is: \[ u_n = a + (n-1)d \]
Key Terminology:
\(a\): The first term.
\(d\): The common difference (found by \(u_{n+1} - u_n\)).
\(n\): The position of the term.
Step-by-Step Example (Finding the 50th Term)
Sequence: 5, 12, 19, 26, ... Find the 50th term.
- Identify \(a\) and \(d\):
\(a = 5\)
\(d = 12 - 5 = 7\) - Identify \(n\):
We want the 50th term, so \(n = 50\). - Substitute into the formula \(u_n = a + (n-1)d\):
\(u_{50} = 5 + (50 - 1)(7)\)
\(u_{50} = 5 + (49)(7)\)
\(u_{50} = 5 + 343 = 348\)
Common Mistake Alert! Always remember the \((n-1)\) in the formula. If you used \(nd\), you would be adding one too many common differences.
2. The Sum of an Arithmetic Series (\(S_n\))
The sum of the first \(n\) terms of an AP is denoted \(S_n\).
Imagine adding the first term (\(a\)) and the last term (\(l\) or \(u_n\)). Then add the second term and the second-to-last term. For any AP, these pairs will always have the same sum!
If there are \(n\) terms, there are \(n/2\) pairs.
The formula for the Sum of the first \(n\) terms of an AP (if you know the last term \(l\)) is: \[ S_n = \frac{n}{2}(a + l) \]
Since we know \(l = u_n = a + (n-1)d\), we can substitute this into the formula to get the second, more common version: \[ S_n = \frac{n}{2}(2a + (n-1)d) \]
Step-by-Step Example (Summing the First 20 Terms)
Find the sum of the first 20 terms of the sequence 3, 7, 11, 15, ...
- Identify \(a\), \(d\), and \(n\):
\(a = 3\)
\(d = 4\)
\(n = 20\) - Use the formula \(S_n = \frac{n}{2}(2a + (n-1)d)\):
\(S_{20} = \frac{20}{2}(2(3) + (20-1)4)\)
\(S_{20} = 10(6 + (19)4)\)
\(S_{20} = 10(6 + 76)\)
\(S_{20} = 10(82) = 820\)
Key Takeaway AP: Arithmetic sequences involve addition of a constant difference \(d\). The sum formula relies on averaging the first and last terms.
Section 3: Geometric Progressions (GP)
A Geometric Progression (GP), or Geometric Sequence, is a sequence where the ratio between consecutive terms is constant. We call this constant ratio the Common Ratio, denoted by \(r\).
Analogy: This is like compound interest or bacterial growth—the increase is based on the current size, not a fixed amount. You are multiplying, not adding.
1. The \(n\)-th Term of a GP
Let the first term be \(a\).
- \(u_1 = a\)
- \(u_2 = a \times r\)
- \(u_3 = (a \times r) \times r = ar^2\)
- \(u_4 = ar^3\)
Again, the power of \(r\) is always one less than the term number \(n\).
The formula for the \(n\)-th term of a GP is: \[ u_n = ar^{n-1} \]
Key Terminology:
\(a\): The first term.
\(r\): The common ratio (found by \(u_{n+1} \div u_n\)).
\(n\): The position of the term.
Step-by-Step Example (Finding the 8th Term)
Sequence: 2, 6, 18, 54, ... Find the 8th term.
- Identify \(a\) and \(r\):
\(a = 2\)
\(r = 6 / 2 = 3\) - Identify \(n\):
We want the 8th term, so \(n = 8\). - Substitute into the formula \(u_n = ar^{n-1}\):
\(u_8 = 2(3)^{8-1}\)
\(u_8 = 2(3)^7\)
\(u_8 = 2(2187) = 4374\)
2. The Sum of a Geometric Series (\(S_n\))
Finding the sum of a GP is more complex algebraically, but the formula is essential.
The formula for the Sum of the first \(n\) terms of a GP is:
Use this form if \(r > 1\): \[ S_n = \frac{a(r^n - 1)}{r - 1} \]
Use this form if \(r < 1\): (This avoids negative numbers in the denominator, making calculations cleaner) \[ S_n = \frac{a(1 - r^n)}{1 - r} \]
Note: These two formulas are algebraically identical; you can use either, but selecting the appropriate one minimizes errors.
Example (Summing the First 6 Terms)
Find the sum of the first 6 terms of the sequence 4, 2, 1, 0.5, ...
- Identify \(a\), \(r\), and \(n\):
\(a = 4\)
\(r = 2 / 4 = 0.5\). Since \(r < 1\), we use the second formula.
\(n = 6\) - Use the formula \(S_n = \frac{a(1 - r^n)}{1 - r}\):
\(S_6 = \frac{4(1 - (0.5)^6)}{1 - 0.5}\)
\(S_6 = \frac{4(1 - 0.015625)}{0.5}\)
\(S_6 = \frac{4(0.984375)}{0.5} = \frac{3.9375}{0.5} = 7.875\)
Did you know? Geometric growth is what makes investments powerful over time. Even a small ratio \(r\) (like 1.05 for 5% interest) leads to huge numbers if \(n\) is large!
Section 4: Sum to Infinity (\(S_\infty\)) of a Geometric Series
Imagine you are jumping halfway toward a wall with every leap. You will keep jumping forever, but you will never actually reach the wall. Your total distance traveled will approach a specific limit.
This idea of "approaching a limit" is the Sum to Infinity. For an infinite Geometric Series to have a finite, measurable sum, it must converge.
1. The Condition for Convergence
A Geometric Series only converges (has a finite sum) if the terms get smaller and smaller, eventually approaching zero. This happens only when the magnitude (absolute value) of the common ratio \(r\) is less than 1.
Convergence Condition: \[ |r| < 1 \quad \text{or} \quad -1 < r < 1 \]
If \(|r| \ge 1\), the terms either stay the same size or grow larger, meaning the sum goes off to infinity (it diverges).
2. The Sum to Infinity Formula
If the series converges, as \(n\) tends to infinity, the term \(r^n\) tends toward zero.
Starting with \(S_n = \frac{a(1 - r^n)}{1 - r}\):
If \(r^n \to 0\), the formula simplifies dramatically.
The formula for the Sum to Infinity is: \[ S_\infty = \frac{a}{1 - r} \]
Example (Calculating \(S_\infty\))
Calculate the sum to infinity for the series 10, 5, 2.5, 1.25, ...
- Identify \(a\) and \(r\):
\(a = 10\)
\(r = 5 / 10 = 0.5\) - Check Convergence:
Since \(|0.5| < 1\), the series converges. - Use the formula \(S_\infty = \frac{a}{1 - r}\):
\(S_\infty = \frac{10}{1 - 0.5}\)
\(S_\infty = \frac{10}{0.5} = 20\)
This means that even if you add infinitely many terms, the total sum will never exceed 20.
Accessibility Tip: Dealing with Simultaneous Equations
Many exam questions will ask you to find \(a\) and \(d\) (for AP) or \(a\) and \(r\) (for GP) given two terms or a term and a sum.
Strategy:
1. Write down two equations based on the information given (e.g., \(u_3 = 10\) becomes \(a + 2d = 10\)).
2. For AP: Use simultaneous linear equations (substitution or elimination).
3. For GP: Use simultaneous equations by division. Divide one equation by the other to cancel \(a\) and solve for \(r\).
Key Takeaway GP & Infinity: Geometric sequences involve multiplication by a common ratio \(r\). The Sum to Infinity only exists if \(r\) is small (between -1 and 1).
You have now mastered the core concepts of Sequences and Series! Keep practicing these formulas, and you’ll find these patterns become second nature.