【Math B】Sequences Master Guide
Hello! In these notes, we will cover "Sequences" from Math B, starting from the very basics. A sequence is essentially a field where you discover the "rules of how numbers are arranged." It might seem difficult at first, but it has the charm of a puzzle.
It may feel tricky initially, but once you memorize the patterns, it becomes a major source of points on the Common Test. Relax, and let's go through it together!
1. The Basics of Sequences: Arithmetic Sequences
Let's start with the simplest type: the arithmetic sequence. This is a sequence where numbers increase or decrease by a constant amount.
Key Points of Arithmetic Sequences
・First term: The very first number in the sequence. Usually written as \( a \).
・Common difference: The "difference" between adjacent numbers. Usually written as \( d \).
・General term: The formula representing the \( n \)-th number, written as \( a_n \).
【Formula: General Term】
\( a_n = a + (n-1)d \)
(Visualizing it: To get to the \( n \)-th term, you just need to jump forward \( n-1 \) times starting from the beginning!)
【Formula: Sum of an Arithmetic Sequence】
\( S_n = \frac{1}{2}n(a + l) \)
(\( n \): number of terms, \( a \): first term, \( l \): last term)
*It’s easy to remember as: "(First + Last) × Number of terms ÷ 2"!
💡 Trivia:
There is a famous episode about the genius mathematician Gauss. When he was a primary school student, his teacher told him to add all numbers from 1 to 100. He instantly solved it by calculating "(1+100) × 100 ÷ 2".
⚠️ Common Mistake:
Many people accidentally use \( n \) instead of \( (n-1) \) in the general term formula. Since "the 1st term involves 0 jumps and the 2nd term involves 1 jump," don't forget to subtract 1!
2. Growing by Multiplication: Geometric Sequences
Next is the geometric sequence, which grows at a constant ratio. This is a sequence where you keep multiplying by the same number.
Key Points of Geometric Sequences
・First term: The first number \( a \)
・Common ratio: The number you multiply by. Usually written as \( r \).
・General term: \( a_n = a r^{n-1} \)
(Visualizing it: To get to the \( n \)-th term, you just need to multiply by the ratio \( n-1 \) times starting from the beginning!)
【Formula: Sum of a Geometric Sequence】
When \( r \neq 1 \):
\( S_n = \frac{a(1 - r^n)}{1 - r} \) or \( \frac{a(r^n - 1)}{r - 1} \)
*Choose the one where the denominator doesn't become negative to reduce calculation errors!
Real-life examples:
Bank interest or the spread of information on social media (one person tells two people, then those two tell two more...) follow this kind of growth pattern. Rapid increase is its signature trait.
3. Mastering the Summation Symbol: Σ (Sigma)
Many students feel intimidated by \( \sum \) (Sigma). But in reality, it's just a "shortcut for addition!"
If you see \( \sum_{k=1}^{n} a_k \), it simply means "Plug 1, 2, 3... into \( k \) in \( a_k \) sequentially, and add them all up until you reach \( n \)!"
Σ Formulas You Must Remember
① \( \sum_{k=1}^{n} c = nc \) (If it's a constant, just multiply by the number of terms!)
② \( \sum_{k=1}^{n} k = \frac{1}{2}n(n+1) \)
③ \( \sum_{k=1}^{n} k^2 = \frac{1}{1}n(n+1)(2n+1) \)
④ \( \sum_{k=1}^{n} k^3 = \{ \frac{1}{2}n(n+1) \}^2 \)
★ Key point:
The formula for \( k^2 \) is quite complex in structure, so write it out repeatedly to build muscle memory. In the Common Test, the "factorization" required after using this formula is often what separates the scores!
4. Arithmetico-Geometric Sequences (Differences)
When you can't see a rule in the sequence itself, try listing the "differences between adjacent terms." Sometimes, a new rule (sequence) appears there. This is called a sequence of differences.
【Formula: General Term using Differences】
When \( n \geqq 2 \):
\( a_n = a_1 + \sum_{k=1}^{n-1} b_k \)
* \( b_k \) is the general term of the sequence of differences.
⚠️ Important Note:
When using this formula, always state that it is "for \( n \geqq 2 \)." Also, make it a habit to plug \( n=1 \) into your final answer to check if it matches the actual first term. This is a common point where points are deducted in written exams!
5. Recurrence Relations: Passing the Baton
A recurrence relation is a "rule to create the next term \( a_{n+1} \) using the previous term \( a_n \)." It's like a recipe for a dish.
3 Basic Patterns
① \( a_{n+1} = a_n + d \) → Arithmetic sequence!
② \( a_{n+1} = r a_n \) → Geometric sequence!
③ \( a_{n+1} = p a_n + q \) → Use the "characteristic equation" to transform it into a geometric sequence form.
(Solve \( \alpha = p \alpha + q \) to turn it into the form \( a_{n+1} - \alpha = p(a_n - \alpha) \))
💡 Tip:
If you get stuck on a recurrence relation, don't panic. Plug in \( n=1, 2, 3 \) to write out the first few numbers. You'll often spot the pattern that way.
6. Mathematical Induction
The last technique is for proofs. It is often compared to "falling dominoes."
Steps for Proof
1. Verify that it holds for \( n=1 \). (Tip the first domino.)
2. Assume it holds for \( n=k \), and use that assumption to show it also holds for \( n=k+1 \). (Prove that if the previous domino falls, the next one will too.)
3. Conclude that it holds for "all natural numbers \( n \)."
Key point:
Write down the "formula for \( n=k+1 \)" as your goal beforehand so you don't get lost in your calculations.
Summary: The Keys to Conquering Sequences
・Memorize the basic arithmetic and geometric formulas perfectly.
・Don't be afraid of Σ; just break it down.
・Identify the type (pattern) of recurrence relations.
・When in doubt, write out the numbers explicitly!
Sequences is a field where you will definitely improve with practice. Don't worry if you make calculation errors at first. Take it one step at a time, steadily. I'm rooting for you!