Sequences, Series and Binomial Theorem

Introduction

Assalamu alaikum, dear students. In your A Level journey, you will find that some of the most elegant ideas in mathematics involve patterns. This topic, covering Sequences, Series, and the Binomial Theorem, is the heart of understanding those patterns. From calculating loan repayments for a new car in Lahore to modelling the population growth of Pakistan, the principles of progressions are everywhere. A sequence is simply a list of numbers in a specific order, while a series is the sum of the terms in that sequence.

The Binomial Theorem, on the other hand, is a remarkable tool of algebra. It provides a shortcut for expanding expressions like (x+y)⁷ without tedious multiplication. For your P3 syllabus, this extends to fractional and negative powers, allowing us to find excellent approximations for complex functions, a technique fundamental in physics and engineering. Mastering these concepts will not only secure you marks in Pure Mathematics but will also build a strong foundation for university-level science and economics. Insha'Allah, by the end of this guide, you will approach these problems with confidence and skill.

Core Theory

Let's break down the core components you need to master for your 9709 examinations.

Arithmetic Progressions (AP)

An Arithmetic Progression is a sequence where each term after the first is found by adding a constant, called the common difference (d), to the previous one.

• First Term: `a` (or `u₁`)
• Common Difference: `d = u₂ - u₁ = u₃ - u₂`

The nth Term (uₙ):

To find any term in the sequence, we use the formula:

`uₙ = a + (n-1)d`

*Example:* For the AP 7, 11, 15, 19, ..., `a=7` and `d=4`. The 20th term is `u₂₀ = 7 + (20-1) * 4 = 7 + 19 * 4 = 7 + 76 = 83`.

The Sum of the First n Terms (Sₙ):

To find the sum of a part of the sequence, use either of these formulae:

1. `Sₙ = n/2 * [2a + (n-1)d]` (Use when you know `a`, `n`, and `d`)
2. `Sₙ = n/2 * (a + l)` (Use when you know the first term `a`, the number of terms `n`, and the last term `l`)

*Example:* The sum of the first 20 terms of the AP above is `S₂₀ = 20/2 * [2(7) + (20-1)4] = 10 * [14 + 76] = 10 * 90 = 900`.

Geometric Progressions (GP)

A Geometric Progression is a sequence where each term after the first is found by multiplying the previous one by a constant, non-zero number called the common ratio (r).

• First Term: `a`
• Common Ratio: `r = u₂/u₁ = u₃/u₂`

The nth Term (uₙ):

`uₙ = arⁿ⁻¹`

*Example:* For the GP 3, 6, 12, 24, ..., `a=3` and `r=2`. The 10th term is `u₁₀ = 3 * 2¹⁰⁻¹ = 3 * 2⁹ = 3 * 512 = 1536`.

The Sum of the First n Terms (Sₙ):

`Sₙ = a(1 - rⁿ) / (1 - r)` or `Sₙ = a(rⁿ - 1) / (r - 1)`

(Beta, use the first form when |r|<1 and the second when |r|>1 to keep the denominator positive and avoid calculation errors).

The Sum to Infinity (S∞):

A very important concept. If the common ratio `r` is between -1 and 1 (i.e., `|r| < 1`), the terms get progressively smaller, and the sum approaches a finite limit. This is called a convergent series.

`S∞ = a / (1 - r)` for `|r| < 1` only.

*Example:* For the GP 8, 4, 2, 1, ..., `a=8` and `r=1/2`. Since `|1/2| < 1`, the sum to infinity exists. `S∞ = 8 / (1 - 1/2) = 8 / (1/2) = 16`.

Binomial Theorem

For Positive Integer `n` (P1 Syllabus):

This gives us the expansion of `(a+b)ⁿ`.

`(a+b)ⁿ = aⁿ + ⁿC₁aⁿ⁻¹b¹ + ⁿC₂aⁿ⁻²b² + ... + bⁿ`

The general term is `ⁿCᵣ aⁿ⁻ᵣ bᵣ`, where `ⁿCᵣ = n! / (r!(n-r)!)`.

*Example:* Find the first 3 terms of `(2 + x)⁴`.

Here `a=2, b=x, n=4`.

Term 1: `⁴C₀ * 2⁴ * x⁰ = 1 * 16 * 1 = 16`

Term 2: `⁴C₁ * 2³ * x¹ = 4 * 8 * x = 32x`

Term 3: `⁴C₂ * 2² * x² = 6 * 4 * x² = 24x²`

So, `(2 + x)⁴ = 16 + 32x + 24x² + ...`

For any Rational `n` (P3 Syllabus):

This is for powers that are negative or fractions. The expansion is infinite and is only valid for a specific range of `x`. The formula is for `(1+x)ⁿ`:

`(1+x)ⁿ = 1 + nx + n(n-1)/2! * x² + n(n-1)(n-2)/3! * x³ + ...`

This expansion is valid only if |x| < 1.

*Example:* Find the first 3 terms of `√(4+x)` and state the range of validity.

First, rearrange into the `(1+...)^n` form:

`√(4+x) = (4(1 + x/4))¹/² = 4¹/² * (1 + x/4)¹/² = 2(1 + x/4)¹/²`

Now expand `(1 + x/4)¹/²` with `n=1/2` and `x` replaced by `x/4`:

`2 * [1 + (1/2)(x/4) + (1/2)(-1/2)/2! * (x/4)² + ...]`

`= 2 * [1 + x/8 - 1/8 * x²/16 + ...]`

`= 2 * [1 + x/8 - x²/128 + ...]`

`= 2 + x/4 - x²/64 + ...`

For validity, the term inside the bracket must have a modulus less than 1: `|x/4| < 1`, which means `|x| < 4`.

Key Definitions

• Sequence: An ordered list of numbers.
• Series: The sum of the terms of a sequence.
• Arithmetic Progression (AP): A sequence with a common difference `d`.
• `uₙ = a + (n-1)d`
• `Sₙ = n/2 * [2a + (n-1)d]`
• Geometric Progression (GP): A sequence with a common ratio `r`.
• `uₙ = arⁿ⁻¹`
• `Sₙ = a(1 - rⁿ) / (1 - r)`
• Sum to Infinity (S∞): The sum of an infinite GP.
• `S∞ = a / (1 - r)`
• Condition for convergence: `|r| < 1`.
• Binomial Theorem (Positive Integer n):
• `(a+b)ⁿ = Σ ⁿCᵣ aⁿ⁻ᵣ bᵣ` from `r=0` to `n`.
• Binomial Theorem (Rational n):
• `(1+x)ⁿ = 1 + nx + n(n-1)/2! * x² + ...`
• Condition for validity: `|x| < 1`.
• Sigma Notation (Σ): A shorthand for summation. e.g., `Σ (2r+1)` from `r=1` to `5` means `(2*1+1) + (2*2+1) + (2*3+1) + (2*4+1) + (2*5+1)`.

Worked Examples (Pakistani Context)

Example 1: Pure Algebraic Problem

The 3rd term of a geometric progression is 4 and the 6th term is -32. Find the first term `a`, the common ratio `r`, and the sum to infinity `S∞`.

Solution:

We are given:

`u₃ = ar² = 4` --- (1)

`u₆ = ar⁵ = -32` --- (2)

To find `r`, we divide equation (2) by equation (1):

`(ar⁵) / (ar²) = -32 / 4`

`r³ = -8`

`r = ³√(-8) = -2`

Now substitute `r = -2` into equation (1) to find `a`:

`a(-2)² = 4`

`a(4) = 4`

`a = 1`

So, the first term is `a = 1` and the common ratio is `r = -2`.

For the sum to infinity, we must check the condition `|r| < 1`.

Here, `|r| = |-2| = 2`. Since `2` is not less than `1`, the series is divergent and the sum to infinity does not exist.

Example 2: PSL Cricket Viewership

The online viewership for the first match of the Pakistan Super League (PSL) was 1.5 million. An aggressive marketing campaign resulted in viewership for each subsequent match being 10% higher than the previous one.

(i) What was the viewership for the 10th match, to 3 significant figures?

(ii) What was the total viewership for the entire tournament of 34 matches, to 3 significant figures?

Solution:

This is a Geometric Progression.

The first term is `a = 1.5` million.

The viewership increases by 10%, so the common ratio `r = 1 + 10/100 = 1.1`.

(i) We need to find the 10th term, `u₁₀`.

`uₙ = arⁿ⁻¹`

`u₁₀ = 1.5 * (1.1)¹⁰⁻¹ = 1.5 * (1.1)⁹`

`u₁₀ ≈ 1.5 * 2.3579 = 3.5369`

To 3 s.f., the viewership for the 10th match was 3.54 million.

(ii) We need to find the sum of the first 34 terms, `S₃₄`.

Since `r > 1`, we use the formula `Sₙ = a(rⁿ - 1) / (r - 1)`.

`S₃₄ = 1.5 * (1.1³⁴ - 1) / (1.1 - 1)`

`S₃₄ = 1.5 * (25.9056 - 1) / 0.1`

`S₃₄ = 15 * (24.9056) ≈ 373.58`

To 3 s.f., the total viewership for the tournament was 374 million.

Exam Technique

• Identify the Progression: The first step in any word problem is to determine if it's an AP (constant amount added/subtracted) or a GP (constant percentage increase/decrease, or multiplied by a factor). Read the question carefully.
• Write Down Formulae: Before you substitute values, always write down the formula you are using (`uₙ = arⁿ⁻¹`, `Sₙ = n/2 * [2a+(n-1)d]`, etc.). This can earn you method marks even if you make a calculation error.
• "Show That" Questions: These questions require you to be meticulous. You cannot skip steps. Start with the given information and show a clear, logical progression to the required result. Do not "fudge" your answer to match the target.
• Binomial Validity: For P3 questions on rational `n` binomial expansions, the condition for validity is almost always worth a mark. For `(a+bx)ⁿ`, the expansion is valid for `|bx/a| < 1`. Always state this and simplify it for `x`.
• Common Mistakes:

1. Confusing AP and GP formulae.
2. Using `n` instead of `n-1` in the term formulae.
3. Forgetting to check `|r| < 1` before calculating a sum to infinity. If the condition is not met, you must state that the sum to infinity does not exist.
4. Errors with signs in the binomial expansion, especially when expanding `(a-bx)ⁿ` or for negative `n`. Be careful.
5. Incorrectly rearranging expressions like `(2+3x)⁻¹` into `2(1 + 3x/2)⁻¹`. Remember to take the constant out with its power: `2⁻¹(1 + 3x/2)⁻¹`.

• Use Your Calculator Wisely: For calculating high powers or `nCr` values, your calculator is your friend. However, show the initial substitution on your script. For example, write `¹⁰C₃` before you write the answer `120`.