Mathematics - Additional (0606) | Series

🚀 Series: Building Blocks of Advanced Maths (0606)

Welcome to the exciting world of Series! This chapter helps you move beyond individual numbers and look at patterns of numbers grouped together. Understanding series is vital because they appear everywhere—from calculating compound interest to modeling population growth and even in advanced physics.

Don't worry if the formulas look long; they are all provided in your exam formula sheet! Our main job here is to understand how and when to use them.

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Section 1: The Binomial Theorem (BT)

What is the Binomial Theorem?

The Binomial Theorem is a powerful shortcut used to expand expressions of the form \((a+b)^n\) where \(n\) is a positive integer, without having to multiply the brackets repeatedly.

Example: Instead of painfully multiplying \((x+2)^{10}\) ten times, the BT gives us a quick way to find any specific term or the entire expansion.

Key Components of the Expansion \((a+b)^n\)

The full expansion looks like this (and is given in the formula list):

\( (a+b)^n = a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + ... + \binom{n}{r}a^{n-r}b^r + ... + b^n \)

The core ingredients you need to identify are:

• \(a\): The first term in the bracket.
• \(b\): The second term in the bracket (make sure to include its sign!).
• \(n\): The power (must be a positive whole number for this syllabus).

Finding the General Term \(T_{r+1}\)

The most common exam questions ask you to find a specific term (like the 4th term, or the term independent of \(x\)). For this, we use the General Term formula.

The \((r+1)^{th}\) term (or \(T_{r+1}\)) is given by:

\( T_{r+1} = \binom{n}{r} a^{n-r} b^r \)

💡 Memory Aid: This term is provided in the formula sheet as \(\binom{n}{r} a^{n-r} b^r\).

Step-by-step for finding a specific term:

1. Identify \(a\), \(b\), and \(n\).
2. Determine \(r\). If you want the \(k^{th}\) term, then \(r = k - 1\). (E.g., for the 5th term, \(r=4\)).
3. Calculate the binomial coefficient \(\binom{n}{r}\) (using the calculator or definition: \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\)).
4. Substitute \(r\), \(n\), \(a\), and \(b\) into the formula and simplify carefully.

🧠 Common Mistake Alert!
If the question asks for the term independent of x, it means the term where the power of \(x\) is zero (i.e., \(x^0\)). You must set the combined power of \(x\) in your general term expression equal to zero and solve for \(r\).

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Section 2: Arithmetic Progressions (AP)

What is an Arithmetic Progression?

An AP is a sequence where the difference between consecutive terms is constant. This constant difference is called the common difference, \(d\).

Analogy: Imagine climbing a ladder where every rung is the same height apart.
Example sequence: 3, 7, 11, 15, 19, ... (Here, \(d=4\)).

Key Definitions

• First Term: \(a\) (or \(u_1\))
• Common Difference: \(d = u_n - u_{n-1}\)
• \(n^{th}\) Term: \(u_n\)
• Sum of \(n\) terms: \(S_n\)

1. The \(n^{th}\) Term (\(u_n\))

To find any term in the sequence, you use the formula (given):

\( u_n = a + (n-1)d \)

Why \((n-1)\)? Because the first term (\(n=1\)) requires zero additions of \(d\). The 5th term requires \(5-1=4\) additions of \(d\).

2. The Sum of the First \(n\) Terms (\(S_n\))

The sum of the first \(n\) terms is also provided in two forms (use whichever is easier based on the information you have):

Form 1 (Using the last term \(l\)):

\( S_n = \frac{n}{2}(a+l) \), where \(l = u_n\).

Form 2 (Using \(a\) and \(d\)):

\( S_n = \frac{n}{2}\{2a+(n-1)d\} \)

Did you know?
The method for finding the sum of an AP was supposedly discovered by the mathematician Gauss when he was just a child! He quickly figured out how to sum the numbers 1 to 100 by pairing them up (1+100, 2+99, etc.).

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Section 3: Geometric Progressions (GP)

What is a Geometric Progression?

A GP is a sequence where the ratio between consecutive terms is constant. This constant factor is called the common ratio, \(r\).

Analogy: Compound interest or a bouncing ball. The increase/decrease is proportional to the current size.
Example sequence: 2, 6, 18, 54, ... (Here, \(r=3\)).

Key Definitions

• First Term: \(a\) (or \(u_1\))
• Common Ratio: \(r = \frac{u_n}{u_{n-1}}\)
• \(n^{th}\) Term: \(u_n\)
• Sum of \(n\) terms: \(S_n\)

1. The \(n^{th}\) Term (\(u_n\))

To find any term in the sequence, you multiply \(a\) by \(r\) a total of \((n-1)\) times (given):

\( u_n = ar^{n-1} \)

2. The Sum of the First \(n\) Terms (\(S_n\))

The formula for the sum of the first \(n\) terms is (given):

\( S_n = \frac{a(1-r^n)}{1-r} \), where \(r \ne 1\).

Note on Notation: Sometimes you might see \( S_n = \frac{a(r^n-1)}{r-1} \). These are mathematically identical. Use the first version (with \(1-r\)) as it's the one usually provided on your formula sheet and works nicely even if \(r\) is less than 1.

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Section 4: Sum to Infinity (\(S_\infty\))

When Does a Sum to Infinity Exist? (Convergence)

This is one of the most important concepts in the Series chapter! For an infinite sequence of terms to have a finite, fixed sum, the terms must get smaller and smaller, approaching zero.

A Geometric Progression only has a Sum to Infinity (\(S_\infty\)) if it is convergent.

The Condition for Convergence is:

\( |r| < 1 \)

This means the common ratio \(r\) must be strictly between -1 and 1 (i.e., \(-1 < r < 1\)).

Analogy: If you have a rubber band and you cut off half of the remaining length repeatedly, you will approach zero, but you will never truly run out of rubber band. The total amount of rubber band you cut off approaches a finite maximum (the original length).

The Formula for Sum to Infinity

If the condition \(|r| < 1\) is met, the Sum to Infinity is (given):

\( S_\infty = \frac{a}{1-r} \)

Why this formula? If \(|r| < 1\), as \(n\) becomes extremely large (approaches infinity), the term \(r^n\) approaches zero. In the full sum formula \( S_n = \frac{a(1-r^n)}{1-r} \), the term \((1-r^n)\) simplifies to \((1-0)\), leaving only \(\frac{a}{1-r}\).

🧐 Explain in Words:
If asked to explain why a GP *does not* have a sum to infinity, state that the common ratio \(r\) is calculated, and since \(|r| \ge 1\), the terms do not approach zero, and thus the series diverges.

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Section 5: Key Differences and Problem Solving

AP vs GP: How to Tell Them Apart

The syllabus requires you to recognize the difference between them. If you misidentify the series type, you will use the wrong formula!

Type	Rule	Test	Variable
Arithmetic (AP)	Constant difference (Addition/Subtraction)	\(u_2 - u_1 = u_3 - u_2\)	Common difference (\(d\))
Geometric (GP)	Constant ratio (Multiplication/Division)	\(\frac{u_2}{u_1} = \frac{u_3}{u_2}\)	Common ratio (\(r\))

Common Problem-Solving Techniques

Many problems involve finding \(a\), \(d\), \(r\), or \(n\) given specific information (like the 3rd term is 10, and the 7th term is 58). These usually lead to simultaneous equations.

Step-by-step approach for finding \(a\) and \(d\) (AP):

1. Translate the given terms into \(u_n\) formulas.
   Example: "The 4th term is 19" becomes \( a + (4-1)d = 19 \implies a + 3d = 19 \).
2. Create two equations with \(a\) and \(d\).
3. Solve the simultaneous equations (usually by elimination for AP).

Step-by-step approach for finding \(a\) and \(r\) (GP):

1. Translate the given terms into \(u_n\) formulas.
   Example: "The 3rd term is 12" becomes \( ar^{3-1} = 12 \implies ar^2 = 12 \).
2. Create two equations with \(a\) and \(r\).
3. Solve the simultaneous equations (usually by division for GP, as this eliminates \(a\)).

⚠️ Don't Forget: When dealing with GP problems that involve the sum to infinity, always check the ratio \(r\). If your calculation yields two possible ratios, e.g., \(r=2\) and \(r=0.5\), only \(r=0.5\) is valid for \(S_\infty\)!