Welcome to Sequences and Series!
In this chapter, we are going to explore the patterns of numbers. Imagine you are stacking oranges in a supermarket or calculating how much money you’ll have in a savings account over ten years. Both of these involve sequences (lists of numbers) and series (adding those numbers together).
Don't worry if you find some of the formulas a bit intimidating at first! We will break them down step-by-step, and you'll soon see that they are just shortcuts to help us find answers quickly.
1. The Basics: What is a Sequence?
A sequence is simply a list of numbers that follows a specific rule. Each number in the list is called a term. We usually use \(u_n\) or \(x_n\) to represent the "nth term" (the term at position \(n\)).
Finding the nth term
Sometimes you are given a formula for the nth term. For example, if \(u_n = 3n + 1\):
To find the 1st term (\(n=1\)): \(u_1 = 3(1) + 1 = 4\)
To find the 2nd term (\(n=2\)): \(u_2 = 3(2) + 1 = 7\)
Recurrence Relations
Sometimes, a term is defined by the term that came before it. This is called a recurrence relation. It looks like this: \(u_{n+1} = f(u_n)\).
Analogy: Think of a ladder. To get to the next rung (\(u_{n+1}\)), you have to start from the rung you are currently on (\(u_n\)).
Example: If \(u_{n+1} = u_n + 5\) and the first term \(u_1 = 2\):
The next term is \(2 + 5 = 7\).
The term after that is \(7 + 5 = 12\).
Types of Sequences
- Increasing sequence: Each term is greater than the one before it (\(u_{n+1} > u_n\)).
- Decreasing sequence: Each term is smaller than the one before it (\(u_{n+1} < u_n\)).
- Periodic sequence: The terms repeat in a cycle (e.g., \(1, 2, 3, 1, 2, 3, ...\)).
Quick Review: \(n\) is always a position (1st, 2nd, 3rd...), so it must always be a positive whole number.
2. Arithmetic Sequences and Series
In an arithmetic sequence, you add (or subtract) the same amount every time. This amount is called the common difference, \(d\).
The nth term of an Arithmetic Sequence
To find any term, we use: \(u_n = a + (n - 1)d\)
Where:
\(a\) = the first term
\(d\) = the common difference
\(n\) = the position of the term
Arithmetic Series (Adding them up)
When we add the terms of a sequence, it becomes a series. We use the symbol \(\Sigma\) (Sigma) to mean "sum of."
The sum of the first \(n\) terms (\(S_n\)) is calculated using:
\(S_n = \frac{n}{2}[2a + (n - 1)d]\)
OR
\(S_n = \frac{n}{2}(a + l)\) where \(l\) is the last term.
Did you know? There is a special shortcut for the sum of the first \(n\) natural numbers (1, 2, 3...):
\(\sum_{r=1}^{n} r = \frac{1}{2}n(n + 1)\)
Common Mistake to Avoid: When using the formula \(u_n = a + (n-1)d\), don't forget the \((n-1)\). If you want the 10th term, you only add the difference 9 times!
Key Takeaway: Arithmetic = Adding. If the pattern involves adding the same number, use the arithmetic formulas.
3. Geometric Sequences and Series
In a geometric sequence, you multiply by the same amount every time. This amount is called the common ratio, \(r\).
The nth term of a Geometric Sequence
The formula is: \(u_n = ar^{n-1}\)
Where:
\(a\) = the first term
\(r\) = the common ratio
Sum of a Finite Geometric Series
To add up the first \(n\) terms:
\(S_n = \frac{a(1 - r^n)}{1 - r}\) (This version is easier if \(r < 1\))
OR
\(S_n = \frac{a(r^n - 1)}{r - 1}\) (This version is easier if \(r > 1\))
Sum to Infinity (\(S_{\infty}\))
Sometimes, if a geometric sequence gets smaller and smaller (converges), we can add up all the terms forever and get a finite number!
This only works if the common ratio \(r\) is between -1 and 1 (written as \(|r| < 1\)).
The formula is: \(S_{\infty} = \frac{a}{1 - r}\)
Analogy: Imagine walking halfway to a wall, then halfway again, then halfway again. You will keep moving forever, but you will never go past the wall. The wall is your "sum to infinity."
Using Logarithms: If a question asks "How many terms are needed for the sum to exceed X?", you will likely need to use logs to solve for \(n\) once you've set up your \(S_n\) formula.
Key Takeaway: Geometric = Growth/Multiplication. If \(|r| < 1\), the series will eventually settle at a total sum (\(S_{\infty}\)).
4. Binomial Expansion
The binomial expansion is a way to expand brackets like \((a + bx)^n\) without having to multiply them out manually many times. For this unit (P2), we only focus on when \(n\) is a positive integer (a whole number).
The Formula
\((a + bx)^n = a^n + \binom{n}{1}a^{n-1}(bx) + \binom{n}{2}a^{n-2}(bx)^2 + ... + (bx)^n\)
Don't let the notation scare you! Here is what you need to know:
1. Powers of \(a\): Start at \(n\) and go down by 1 each time.
2. Powers of \(bx\): Start at 0 and go up by 1 each time.
3. The Coefficients: The numbers \(\binom{n}{r}\) (also written as \(^nC_r\)) can be found using the button on your calculator or by using the formula \(\frac{n!}{r!(n-r)!}\).
Memory Aid: In every term of the expansion, the two powers must always add up to \(n\). For example, in \((a + b)^5\), one term might be something multiplied by \(a^3b^2\). Notice that \(3 + 2 = 5\).
Step-by-Step for Binomial Expansion:
- Identify your \(a\), your \(bx\), and your \(n\).
- Write out the terms using the \(\binom{n}{r}\) formula.
- Carefully calculate the powers of the numbers (especially if there is a coefficient inside the bracket like \(2x\)).
- Simplify each term at the end.
Common Mistake: Forgetting to square or cube the coefficient of \(x\). If your term is \((3x)^2\), it becomes \(9x^2\), not \(3x^2\)!
Key Takeaway: Binomial expansion is just a structured way to multiply out brackets. Keep your work organized in columns to avoid losing track of minus signs!
Final Encouragement
Sequences and series are all about spotting the "rule" of the pattern. Once you identify if a pattern is Arithmetic (adding) or Geometric (multiplying), you just need to pick the right formula from your toolkit. Practice identifying \(a\), \(d\), and \(r\) first, and the rest will fall into place!