Welcome to Sequences and Series!
Hello future mathematician! This chapter, Sequences and Series, is a fundamental building block in Pure Mathematics 2. It’s all about spotting patterns and creating rules to describe them. Don't worry if it seems abstract; we use sequences every day, from calculating loan interest to predicting population growth!
In this unit, we will unlock the secrets of two major types of patterns: Arithmetic Progressions (AP) and Geometric Progressions (GP). By the end, you'll be able to calculate the 100th term of a sequence or sum up millions of terms almost instantly!
Section 1: Understanding Sequences and Series
1.1 What are Sequences and Series?
A Sequence is simply a list of numbers arranged in a specific order, following a rule.
Example: 2, 4, 6, 8, 10, ... (The rule is "add 2").
A Series is the sum of the terms in a sequence.
Example: 2 + 4 + 6 + 8 + 10
Key Terminology and Notation
- Term (\(u_n\)): Each number in the sequence is called a term. \(u_1\) is the first term, \(u_2\) is the second term, and \(u_n\) is the \(n\)-th term.
- General Term (\(u_n\)): This is the formula or rule that allows you to calculate any term in the sequence just by knowing its position \(n\).
Quick Review: Sequences are lists, Series are sums.
1.2 Defining Sequences
Sequences can be defined in two main ways:
- By Position (Explicit Formula): This gives the rule directly in terms of \(n\).
Example: If the sequence is defined by \(u_n = 3n - 1\). To find the 5th term (u_5), you just plug in \(n=5\): \(u_5 = 3(5) - 1 = 14\). - By Recurrence Relation: This defines a term based on the previous term(s). You always need to know the first term(s) to start.
Example: \(u_{n+1} = 2u_n + 3\), with \(u_1 = 1\).- \(u_1 = 1\)
- \(u_2 = 2u_1 + 3 = 2(1) + 3 = 5\)
- \(u_3 = 2u_2 + 3 = 2(5) + 3 = 13\)
1.3 Sigma Notation (\(\Sigma\))
Sigma notation is a fancy, mathematical shorthand for writing the sum of a series.
The symbol \(\mathbf{\Sigma}\) (the Greek letter Sigma) means "sum".
A general series sum looks like this:
- \(r=1\): This is the lower limit—where you start summing (usually \(r=1\) for the first term).
- \(n\): This is the upper limit—where you stop summing.
- \(u_r\): This is the formula for the terms you are adding together.
Example: Find the sum of the first 4 terms of the sequence \(u_r = 2r\).
This means: \((2 \times 1) + (2 \times 2) + (2 \times 3) + (2 \times 4) = 2 + 4 + 6 + 8 = 20\)
Section 2: Arithmetic Progressions (AP)
An Arithmetic Progression (AP) is a sequence where the difference between consecutive terms is constant. This constant difference is called the common difference, \(d\).
Analogy: Think of receiving a fixed pay rise every year, or climbing a ladder where every rung is the same distance apart.
2.1 The \(n\)-th Term of an AP (\(u_n\))
To find any term in an AP, you need two things: the first term (\(a\)) and the common difference (\(d\)).
Let's look at the terms:
- \(u_1 = a\)
- \(u_2 = a + d\)
- \(u_3 = a + 2d\)
- \(u_4 = a + 3d\)
Notice that the number of times you add \(d\) is always one less than the term number (\(n\)).
The Formula for the \(n\)-th term:
Memory Aid: You start at 'a', and you take 'n' steps, but since you are already at the first term, you only need (n-1) jumps of size 'd'.
2.2 The Sum of an Arithmetic Series (\(S_n\))
The sum of the first \(n\) terms of an AP is denoted by \(S_n\).
We have two main formulas for the sum, depending on what information you are given:
Formula 1 (When you know \(a\), \(n\), and \(d\)):
Formula 2 (When you know \(a\), \(n\), and the last term \(l\)):
Since \(l = u_n = a + (n-1)d\), we can substitute \(l\) into the formula above:
Did you know? This formula was reputedly discovered by mathematician Carl Friedrich Gauss when he was just a child. He realized you could pair the first and last terms, the second and second-to-last, etc., and each pair would sum to the same total.
Common Mistake to Avoid in AP:
Always remember the bracket \((n-1)\) in the \(u_n\) formula. A very common error is writing \(u_n = a + nd\). If you do this, your \(u_1\) term would incorrectly be \(a+d\).
Key Takeaway (AP): Arithmetic means adding a constant difference (\(d\)). Use \(u_n\) for finding a specific term and \(S_n\) for finding the total sum.
Section 3: Geometric Progressions (GP)
A Geometric Progression (GP) is a sequence where the ratio between consecutive terms is constant. This constant multiplier is called the common ratio, \(r\).
Analogy: Think of compound interest or a viral spread, where the growth rate depends on the current size.
3.1 The \(n\)-th Term of a GP (\(u_n\))
To find any term in a GP, you need the first term (\(a\)) and the common ratio (\(r\)).
Let's look at the terms:
- \(u_1 = a\)
- \(u_2 = a \times r = ar\)
- \(u_3 = ar \times r = ar^2\)
- \(u_4 = ar^2 \times r = ar^3\)
Notice that the power of \(r\) is always one less than the term number (\(n\)).
The Formula for the \(n\)-th term:
Step-by-Step for Finding \(r\):
If you are given two consecutive terms, say \(u_5\) and \(u_4\), you find \(r\) by dividing:
\(r = \frac{u_{n}}{u_{n-1}}\)
Example: If the sequence is 5, 10, 20, ... then \(r = 10/5 = 2\).
3.2 The Sum of a Geometric Series (\(S_n\))
The sum of the first \(n\) terms of a GP is denoted by \(S_n\). We have two standard versions of the formula, which are mathematically identical, but one is easier to calculate depending on \(r\).
The Formula for the Sum of \(n\) terms:
OR
Don't worry about memorizing which to use when. As long as you stick to one formula, you will get the correct answer. However, using the first version when \(r\) is positive and less than 1 (like \(r=0.5\)) keeps the denominator positive and avoids negative signs everywhere.
Key Takeaway (GP): Geometric means multiplying by a constant ratio (\(r\)). This leads to rapid growth or decay.
Section 4: Sum to Infinity (\(S_\infty\))
Imagine adding an infinite number of terms. Does the sum get infinitely large? Not always!
For a Geometric Series, if the common ratio \(r\) is small enough, the terms eventually become so close to zero that they barely contribute to the sum. When this happens, the series converges to a finite value.
Analogy: A bouncing ball. Each bounce is a fraction of the height of the previous one. If you sum all the heights of the bounces (an infinite number), the total distance traveled is finite because the bounces get infinitesimally small.
4.1 The Condition for Convergence
A Geometric Series only converges (has a Sum to Infinity) if the common ratio \(r\) lies between -1 and 1.
If \(|r| \geq 1\), the terms either stay the same size or get larger (diverge), and the sum tends towards infinity (or oscillates). No \(S_\infty\) can be calculated.
4.2 The Sum to Infinity Formula
If the condition \(|r| < 1\) is met, we use the formula:
Why does this work? When \(|r| < 1\) and \(n\) becomes very large (tends to infinity), the term \(r^n\) in the \(S_n\) formula tends towards zero. If \(r^n \to 0\), the \(S_n\) formula simplifies from \(\frac{a(1-r^n)}{1-r}\) to \(\frac{a(1-0)}{1-r}\).
Tip for Struggling Students:
When solving problems involving \(S_\infty\), always write down the convergence condition (\(|r|<1\)) first. If the problem asks you to find the possible values of \(x\) for convergence, this is the inequality you must solve!
Quick Review: Essential P2 Formulas
Keep these formulas handy! You must be able to recognize which sequence type you are dealing with before choosing the formula.
Arithmetic Progressions (AP)
- \(n\)-th term: \(u_n = a + (n-1)d\)
- Sum of \(n\) terms: \(S_n = \frac{n}{2} [2a + (n-1)d]\) OR \(S_n = \frac{n}{2} (a + l)\)
Geometric Progressions (GP)
- \(n\)-th term: \(u_n = ar^{n-1}\)
- Sum of \(n\) terms: \(S_n = \frac{a(1-r^n)}{1-r}\)
- Sum to Infinity: \(S_{\infty} = \frac{a}{1-r}\) (Only if \(-1 < r < 1\))
You have successfully navigated the world of patterns! Practice applying these formulas to various contexts, and you will master this chapter!