Series: Unlocking the Patterns of Numbers
Hello future mathematician! This chapter, Series, is all about patterns. Mathematics is built on patterns, and sequences and series give us powerful tools to predict what comes next, find totals quickly, and model real-world growth and decay.
Don't worry if this seems tricky at first. We will break down complex formulas into simple steps. By the end of this chapter, you’ll be an expert at handling specific types of patterns called Arithmetic Progressions (AP) and Geometric Progressions (GP). Let's get started!
1. Sequences vs. Series: Getting the Terms Right
1.1 What's the Difference?
It’s easy to confuse these two terms, but the difference is fundamental:
- Sequence: A list of numbers following a specific rule or pattern. We separate the terms with commas.
- Series: The sum of the terms in a sequence. We connect the terms with plus signs (+).
Example:
Sequence: 1, 3, 5, 7, 9
Series: \(1 + 3 + 5 + 7 + 9\)
Key Takeaway: A Sequence is the list; a Series is the sum of that list.
2. Arithmetic Progressions (AP)
An Arithmetic Progression (AP) is a sequence where you get from one term to the next by adding a fixed number. This fixed number is called the common difference, denoted by \(d\).
Analogy: Think of climbing a staircase where every step is exactly the same height (\(d\)).
2.1 The \(n^{th}\) Term of an AP
If the first term is \(a\) (or \(u_1\)), the formula for the \(n^{th}\) term, \(u_n\), is: \[u_n = a + (n - 1)d\]
Step-by-Step: Finding a Specific Term
Example: Find the 15th term of the AP: 3, 7, 11, 15, ...
- Identify \(a\) (First Term): \(a = 3\)
- Identify \(d\) (Common Difference): \(d = 7 - 3 = 4\)
- Identify \(n\) (Term Number): \(n = 15\)
- Substitute into the formula \(u_n = a + (n - 1)d\):
\(u_{15} = 3 + (15 - 1)4\) - Calculate:
\(u_{15} = 3 + (14)4\)
\(u_{15} = 3 + 56 = 59\)
2.2 Sum of an AP (\(S_n\))
The sum of the first \(n\) terms of an AP is denoted by \(S_n\). We have two formulas you can use, depending on what information you have:
Formula 1 (Using First and Last Term)
If you know the first term (\(a\)) and the last term (\(l\) or \(u_n\)): \[S_n = \frac{n}{2}(a + l)\]
Formula 2 (Using Common Difference)
If you only know \(a\), \(d\), and \(n\): \[S_n = \frac{n}{2}[2a + (n - 1)d]\]
Memory Aid: The formula for the sum uses the average term \(\frac{(a+l)}{2}\) multiplied by the number of terms \(n\).
Common Mistake: When using Formula 2, remember that the \(2a\) is separate from the \((n-1)d\) part inside the main bracket. Calculate \((n-1)d\) first, then add \(2a\).
Quick Review: AP Formulas
- \(n^{th}\) Term: \(u_n = a + (n - 1)d\)
- Sum: \(S_n = \frac{n}{2}(a + l)\) or \(S_n = \frac{n}{2}[2a + (n - 1)d]\)
3. Geometric Progressions (GP)
A Geometric Progression (GP) is a sequence where you get from one term to the next by multiplying by a fixed number. This fixed number is called the common ratio, denoted by \(r\).
Analogy: Think of compound interest or population growth (or decay), where the size of the increase depends on the current size.
3.1 The \(n^{th}\) Term of a GP
If the first term is \(a\), the formula for the \(n^{th}\) term, \(u_n\), is: \[u_n = ar^{n-1}\]
Example: Find the 7th term of the GP: 2, 6, 18, ...
- Identify \(a\): \(a = 2\)
- Identify \(r\): \(r = 6 \div 2 = 3\)
- Identify \(n\): \(n = 7\)
- Substitute into the formula \(u_n = ar^{n-1}\):
\(u_7 = 2 \times 3^{7-1}\) - Calculate:
\(u_7 = 2 \times 3^6\)
\(u_7 = 2 \times 729 = 1458\)
3.2 Sum of a GP (\(S_n\))
The sum of the first \(n\) terms of a GP is given by two equivalent formulas. Use the one that keeps the denominator positive, as it usually makes calculation easier.
Formula Set:
\[S_n = \frac{a(r^n - 1)}{r - 1} \quad \text{ (Use when } r > 1 \text{)}\]
\[S_n = \frac{a(1 - r^n)}{1 - r} \quad \text{ (Use when } r < 1 \text{)}\]
Did you know? These formulas come from a clever trick: multiplying the series by \(r\) and subtracting it from the original series, which cancels out most of the terms!
4. Sum to Infinity (\(S_{\infty}\))
This is one of the most powerful and interesting concepts in series! The Sum to Infinity, \(S_{\infty}\), is the total sum you get if you keep adding terms in a GP forever.
4.1 When Does \(S_{\infty}\) Exist? (Convergence)
A sum to infinity only exists if the terms are getting smaller and smaller, rapidly approaching zero.
This happens only when the common ratio \(r\) is between -1 and 1.
Condition for Convergence: \[|r| < 1 \quad \text{or} \quad -1 < r < 1\]
If \(|r| \geq 1\), the series diverges—the sum grows infinitely large, and \(S_{\infty}\) cannot be calculated.
4.2 The Sum to Infinity Formula
If the convergence condition is met, the formula is beautifully simple: \[S_{\infty} = \frac{a}{1 - r}\]
Step-by-Step Example: Calculate \(S_{\infty}\) for the series \(6 + 3 + 1.5 + 0.75 + ...\)
- Identify \(a\): \(a = 6\)
- Identify \(r\): \(r = 3 \div 6 = 0.5\)
- Check Condition: Since \(|0.5| < 1\), \(S_{\infty}\) exists.
- Substitute:
\(S_{\infty} = \frac{6}{1 - 0.5}\) - Calculate:
\(S_{\infty} = \frac{6}{0.5} = 12\)
Crucial Concept Check
Before calculating \(S_{\infty}\), ALWAYS check the value of \(r\). If \(r=2\), the series diverges, and the answer is not defined. If the question asks for \(S_{\infty}\), you must state if it converges or diverges based on \(r\).
5. Advanced Applications and Problem Solving
5.1 Solving for Unknown Terms
In Further Pure Maths, you often need to find the terms \(a\), \(d\), \(r\), or \(n\) when given other information.
Working with Ratios (GP)
If you are given two terms of a GP, divide them to eliminate \(a\).
Example: If \(u_3 = 45\) and \(u_6 = 1215\).
\(u_3 = ar^2 = 45\)
\(u_6 = ar^5 = 1215\)
Divide \((u_6 / u_3)\):
\[\frac{ar^5}{ar^2} = \frac{1215}{45} \implies r^3 = 27 \implies r = 3\]
Finding the Number of Terms (\(n\))
If you are solving for \(n\) in a GP, you must use logarithms since \(n\) is in the exponent \((n-1)\).
Example: Find \(n\) if \(u_n = 2000\), \(a=5\), \(r=2\).
\(5 \times 2^{n-1} = 2000\)
\(2^{n-1} = 400\)
Take \(\log\) of both sides:
\[\log(2^{n-1}) = \log(400)\]
\[(n-1) \log(2) = \log(400)\]
\[n-1 = \frac{\log(400)}{\log(2)}\]
(Calculation gives \(n-1 \approx 8.64\). Since \(n\) must be an integer, you would need to check the 8th and 9th terms.)
5.2 When Terms are Consecutive (Means)
If three numbers \(x, y, z\) are consecutive terms in a progression:
- If AP: The middle term is the average: \(y = \frac{x+z}{2}\), or \(2y = x+z\).
- If GP: The middle term squared is the product of the others: \(y^2 = xz\). (This is the Geometric Mean).
Key Takeaway: Series problems often require simultaneous equations or logarithmic manipulation. Master the basic formulas first, and the harder problems will follow.
Keep practicing these formulas and techniques. You have built a strong foundation in patterns that will serve you well throughout Further Pure Mathematics!