Lesson: Sequences and Series
Hello to all you future TCAS candidates! Welcome to the "Sequences and Series" lesson, one of the most frequently tested topics in the A-Level Applied Mathematics 1 exam, under the Number and Algebra strand.
If you've ever noticed "patterns" in your daily life, such as saving an extra 10 baht every day or bacteria doubling their population every hour, that is exactly the heart of this topic! If the formulas seem overwhelming at first, don't worry—we’ll break them down together, step by step.
1. What are Sequences?
A sequence is an ordered set of numbers that follows a specific rule. We represent each term in the sequence with the notation \(a_1, a_2, a_3, ..., a_n\)
- \(a_1\) is the 1st term (the starting number).
- \(a_n\) is the \(n\)-th term or the "general term" (the value at any position we want to find).
1.1 Arithmetic Sequence
This is a sequence where each subsequent term is found by "adding" the same constant. We call this constant the Common Difference, represented by the symbol \(d\).
General term formula: \(a_n = a_1 + (n-1)d\)
Key Point: An easy way to find \(d\) is to take "a later term minus an earlier term" (e.g., \(a_2 - a_1\)).
Real-life example: Climbing stairs with equal height steps, or if Phon-sawan saves 5 baht on the first day, 10 baht the next, and 15 baht the following (here, \(d = 5\)).
1.2 Geometric Sequence
This is a sequence where each subsequent term is found by "multiplying" by the same constant. We call this constant the Common Ratio, represented by the symbol \(r\).
General term formula: \(a_n = a_1 \cdot r^{n-1}\)
Key Point: The way to find \(r\) is to take "a later term divided by an earlier term" (e.g., \(\frac{a_2}{a_1}\)).
Real-life example: The spread of a rumor (1 person tells 3 people, those 3 each tell 3 more, resulting in 9 people...) in this case, \(r = 3\).
A quick recap so you don't forget:
Arithmetic: Focuses on addition (\(d\))
Geometric: Focuses on multiplication (\(r\))
2. What are Series?
If a "sequence" is just listing numbers in order, a "series" is simply the "sum" of those numbers! We use the symbol \(S_n\) to represent the sum of the first \(n\) terms.
2.1 Arithmetic Series
When we sum an arithmetic sequence, we have two main formulas:
- When you know the last term (\(a_n\)): \(S_n = \frac{n}{2}(a_1 + a_n)\)
- When you don't know the last term: \(S_n = \frac{n}{2}[2a_1 + (n-1)d]\)
Memorization tip: The first formula is like "averaging the first and last terms, then multiplying by the number of terms."
2.2 Geometric Series
This is the sum of a geometric sequence. The formula is:
\(S_n = \frac{a_1(1-r^n)}{1-r}\) (where \(r \neq 1\))
Caution: If you encounter a problem where \(r = 1\), it means every number in the sequence is the same. The sum is simply \(n \times a_1\).
3. Summation Notation (\(\sum\) - Sigma)
The \(\sum\) (Sigma) symbol is a compact, elegant way to write sums and save space.
\(\sum_{i=1}^{n} a_i\) means to take the values of \(a_i\) and add them up from \(i=1\) to \(i=n\).
Formulas you must memorize (used very often):
1. \(\sum_{i=1}^{n} i = \frac{n(n+1)}{2}\) (sum of consecutive integers from 1 to \(n\))
2. \(\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}\)
3. \(\sum_{i=1}^{n} i^3 = [\frac{n(n+1)}{2}]^2\)
Did you know? The formula \(\sum_{i=1}^{n} c = n \cdot c\) (where \(c\) is a constant) means that if we add the same number \(n\) times, it's the same as multiplication!
4. Infinite Series
In the A-Level 1 curriculum, we focus on Infinite Geometric Series, which involves summing terms indefinitely.
This series can be solved (it converges) only if \(-1 < r < 1\) (or written as \(|r| < 1\)).
Infinite sum formula: \(S_{\infty} = \frac{a_1}{1-r}\)
Caution: If \(|r| \geq 1\), for example, \(r = 2\), the sum will grow indefinitely and cannot be calculated. This is called a "Divergent Series".
Common Mistakes
1. Confusing \(n\) with \(a_n\): Remember that \(n\) is the "position number" (always a positive integer), whereas \(a_n\) is the "value of the term at that position."
2. Forgetting parentheses: Especially when using the \(S_n\) formula for arithmetic series, forgetting to put parentheses around \((n-1)d\) often leads to calculation errors.
3. Signs of \(d\) and \(r\): If the sequence is decreasing, \(d\) is negative. If the sequence alternates between positive and negative signs, \(r\) is negative. Don't forget your signs!
Key Takeaways for the Exam
Step 1: Read the problem and check if it's "Arithmetic" (adding) or "Geometric" (multiplying).
Step 2: Clearly identify the given variables (\(a_1, n, d\), or \(r\)).
Step 3: Choose the appropriate formula and substitute the values carefully.
You've got this! Once you learn to spot the "pattern," no matter how the numbers change, you'll be able to solve it for sure! Practice often, and you'll get better and better!