Hello Grade 12 students! Welcome to the world of "Sequences and Series."
If you've ever noticed the patterns on a seashell, your daily savings, or the long-term behaviour of an investment over time, you are encountering what we call Sequences and Series! In Grade 12, we build on the foundations of patterns to explore what happens when sequences and series continue indefinitely toward infinity. If it looks challenging at first, don't worry! We'll walk through it together, step by step.
1. Sequences: Finite and Infinite
A sequence is a function whose domain is a set of positive integers written in a specific order based on a rule. Each individual number is called a term:
- The 1st term is denoted as \(a_1\)
- The 2nd term is denoted as \(a_2\)
- The \(n\)-th term (the general term) is denoted as \(a_n\)
A sequence that ends at a specific term \(a_n\) is a finite sequence, whereas a sequence that continues endlessly \(a_1, a_2, a_3, \dots, a_n, \dots\) is an infinite sequence.
1.1 Arithmetic Sequence
Take a look at this: 2, 5, 8, 11, ... Notice that it adds 3 every single time! A sequence that increases or decreases by adding a constant value is called an arithmetic sequence.
Key Points:
- The constant value added is called the common difference, represented by the symbol \(d\).
- The formula to find the \(n\)-th term is: \(a_n = a_1 + (n-1)d\)
Example: For 5, 7, 9, ... to find the 10th term (\(a_{10}\)):
- \(a_1 = 5\)
- \(d = 2\) (since \(7 - 5 = 2\))
- Substitute into the formula: \(a_{10} = 5 + (10-1)(2) = 5 + 18 = 23\)
1.2 Geometric Sequence
Now look at this: 2, 4, 8, 16, ... It does not add the same amount, but it multiplies by 2 each time! A sequence that changes by multiplying by a constant value is called a geometric sequence.
Key Points:
- The constant multiplier is called the common ratio, represented by the symbol \(r\).
- The formula to find the \(n\)-th term is: \(a_n = a_1 \cdot r^{n-1}\)
- You can find \(r\) by dividing any term by its preceding term: \(r = \frac{a_{n}}{a_{n-1}}\), such as \(r = \frac{a_2}{a_1}\).
2. Limits of Infinite Sequences: Convergence and Divergence
When an infinite sequence continues without end, we examine what happens to \(a_n\) as \(n\) grows larger and larger without bound (\(n \to \infty\)). This value is called the limit of the sequence, written as:
\(\lim_{n \to \infty} a_n = L\)
- Convergent Sequence (ลำดับลู่เข้า): If \(\lim_{n \to \infty} a_n\) exists and approaches a single real number \(L\). For example, for \(a_n = \frac{1}{n}\), as \(n\) gets infinitely large, \(a_n\) approaches 0, so the sequence converges to 0.
- Divergent Sequence (ลำดับลู่ออก): If the sequence does not approach any finite real number (it grows infinitely large toward \(\infty\) or \(-\infty\), or oscillates back and forth). For example, \(a_n = 2n\) or \(a_n = (-1)^n\) are divergent sequences.
3. Series: Finite and Infinite Summations
When we add the terms of a sequence, we form a series. The sum of the first \(n\) terms is called the \(n\)-th partial sum, denoted by \(S_n\):
\(S_n = a_1 + a_2 + a_3 + \dots + a_n\)
3.1 Arithmetic Series
When adding terms of an arithmetic sequence, use either formula:
1. When the last term (\(a_n\)) is known: \(S_n = \frac{n}{2}(a_1 + a_n)\)
2. When the last term is unknown: \(S_n = \frac{n}{2}[2a_1 + (n-1)d]\)
3.2 Geometric Series (Finite Sum)
When adding the first \(n\) terms of a geometric sequence:
\(S_n = \frac{a_1(1-r^n)}{1-r}\) (where \(r \neq 1\))
3.3 Infinite Geometric Series and Sum to Infinity
An infinite series is written as \(a_1 + a_2 + a_3 + \dots\). We define its sum \(S\) as the limit of partial sums:
\(S = \lim_{n \to \infty} S_n\)
For an infinite geometric series:
- If \(|r| < 1\) (i.e. \(-1 < r < 1\)), as \(n \to \infty\), the term \(r^n \to 0\). Therefore, the series is convergent and its sum to infinity is given by the formula:
\(S = \frac{a_1}{1-r}\)
- If \(|r| \ge 1\), the series is divergent, meaning it does not have a finite sum.
Example: Find the sum of the infinite series \(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots\)
Here, \(a_1 = 1\) and \(r = \frac{1}{2}\). Since \(|r| = \frac{1}{2} < 1\), the series converges:
\(S = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2\)
4. Common Mistakes
- Using \(S = \frac{a_1}{1-r}\) when \(|r| \ge 1\): The infinite sum formula is only valid when \(|r| < 1\). If \(r = 2\), the terms grow without bound and the series diverges.
- Confusing the \(n-1\) index: In the formulas \(a_n = a_1 + (n-1)d\) and \(a_n = a_1 r^{n-1}\), remember that the first term starts at exponent or multiplier 0, which is why we subtract 1 from \(n\).
- Confusing \(a_n\) and \(S_n\): Remember that \(a_n\) is the specific single term at position \(n\), whereas \(S_n\) represents the sum of all terms from \(a_1\) up to \(a_n\).
5. Final Wrap-up: Tips for Success
1. Classify the sequence: Is it arithmetic (common difference \(d\)) or geometric (common ratio \(r\))?
2. Check finite vs infinite: Are you asked for a finite term/sum (\(a_n\), \(S_n\)) or an infinite limit/sum (\(\lim_{n \to \infty} a_n\), \(S\))?
3. Check convergence condition: For infinite geometric series, always verify whether \(|r| < 1\) before applying the sum formula.
4. Substitute carefully: Watch your signs when \(d\) or \(r\) is negative!