Introduction to Arithmetic Sequences

Welcome! In this chapter, we are exploring one of the most fundamental patterns in mathematics: the Arithmetic Sequence. Think of an arithmetic sequence as a set of stepping stones that are all spaced exactly the same distance apart. Whether you are counting your savings, predicting the growth of a plant, or even just climbing a staircase, you are using the logic of arithmetic sequences!

Don't worry if you find sequences a bit intimidating at first. By the end of these notes, you'll see that it all boils down to just two main numbers: where you start and how much you jump by.

1. What is an Arithmetic Sequence?

An Arithmetic Sequence (sometimes called an Arithmetic Progression or AP) is a list of numbers where the difference between one term and the next is always the same constant.

Key Terms:
1. First Term (\(a\)): This is simply the very first number in your list.
2. Common Difference (\(d\)): This is the "jump" between numbers. You find it by subtracting any term from the one that follows it (\(u_2 - u_1\)).

Example: 5, 8, 11, 14, 17...
Here, the first term \(a = 5\).
The common difference \(d = 3\) (because 8 - 5 = 3, and 11 - 8 = 3).

Did you know?

The common difference \(d\) doesn't have to be a positive whole number! It can be negative (meaning the sequence goes down, like 10, 7, 4...) or even a fraction or decimal.

Common Mistake to Avoid:
Always check that the difference is the same throughout the entire sequence. If it changes, it's not arithmetic!

Quick Review:
- If \(d > 0\), the sequence is increasing.
- If \(d < 0\), the sequence is decreasing.
- If \(d = 0\), the sequence stays the same!

2. Finding Any Term: The \(n\text{th}\) Term Formula

What if you have the sequence 5, 8, 11... and you want to find the 100th number? You wouldn't want to add 3 a hundred times! Luckily, we have a formula.

The \(n\text{th}\) term (usually written as \(u_n\)) is given by:
\(u_n = a + (n - 1)d\)

Breaking down the formula:

  • \(u_n\): The value of the term at position \(n\).
  • \(a\): Where you started.
  • \(n - 1\): Why minus one? Because to get to the 2nd term, you only jump once. To get to the 3rd term, you jump twice. You always jump one less than the position number!
  • \(d\): The size of each jump.

Step-by-Step Example:

Find the 20th term of the sequence: 10, 14, 18, 22...
1. Identify \(a\): \(a = 10\).
2. Identify \(d\): \(14 - 10 = 4\), so \(d = 4\).
3. Identify \(n\): We want the 20th term, so \(n = 20\).
4. Plug into the formula: \(u_{20} = 10 + (20 - 1) \times 4\).
5. Calculate: \(u_{20} = 10 + (19 \times 4) = 10 + 76 = 86\).
The 20th term is 86.

Section Summary: To find any specific term, start at the first term and add the common difference \(n-1\) times.

3. Arithmetic Series: Summing it all up

A Series is what you get when you add the terms of a sequence together. In your exam, the sum of the first \(n\) terms is written as \(S_n\).

There are two formulas you can use, depending on what information you have:

Formula A: If you know the Last Term (\(l\))

\(S_n = \frac{n}{2}(a + l)\)

Analogy: This is like finding the average of the first and last numbers and multiplying by how many numbers there are. It's very quick!

Formula B: If you don't know the Last Term

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

This formula is just Formula A with the \(n\text{th}\) term formula substituted in for \(l\). Use this when you only have \(a\), \(d\), and \(n\).

The Legend of Young Gauss

There is a famous story that a math teacher once asked 7-year-old Carl Friedrich Gauss to add up all the numbers from 1 to 100 to keep him busy. Gauss finished in seconds! He realized that \(1 + 100 = 101\), \(2 + 99 = 101\), \(3 + 98 = 101\), and so on. He had 50 pairs of 101, which is 5050. This is exactly what Formula A does!

Key Takeaway: If a question asks for the "sum of the first... terms," look for the \(S_n\) formulas.

4. Sigma Notation (\(\sum\))

Sometimes, the exam will use a shorthand symbol called Sigma (\(\sum\)) to ask for a sum. It looks scary, but it's just instructions!

\(\sum_{r=1}^{n} (u_r)\)

  • The bottom number (\(r=1\)) tells you which term to start with.
  • The top number (\(n\)) tells you which term to end with.
  • The expression (\(u_r\)) tells you the rule for the sequence.

Example: \(\sum_{r=1}^{4} (2r + 1)\)
Term 1 (\(r=1\)): \(2(1)+1 = 3\)
Term 2 (\(r=2\)): \(2(2)+1 = 5\)
Term 3 (\(r=3\)): \(2(3)+1 = 7\)
Term 4 (\(r=4\)): \(2(4)+1 = 9\)
The sum is \(3 + 5 + 7 + 9 = 24\).

5. Real-World Modeling

Arithmetic sequences are perfect for Modelling situations where something increases or decreases by a set amount over time.

Examples include:
- Simple Interest: If you earn £10 interest every year, your total interest follows an arithmetic sequence.
- Taxi Fares: A flat start fee plus a fixed amount per mile.
- Physical Training: Increasing your running distance by 0.5km every week.

Encouraging Phrase: When tackling "word problems," always start by writing down what \(a\) (the starting value) and \(d\) (the rate of change) are. Once you have those, the rest is just using your formulas!

Chapter Summary Review

Checklist:
- Sequence = A list of numbers.
- Series = The sum of those numbers.
- \(a\) = First term; \(d\) = Common difference.
- \(n\text{th}\) term: \(u_n = a + (n-1)d\).
- Sum of \(n\) terms: \(S_n = \frac{n}{2}(a + l)\) or \(S_n = \frac{n}{2}[2a + (n-1)d]\).
- Sigma (\(\sum\)) is just a fancy way to write "add these up."

Don't worry if this seems tricky at first! The best way to master this is to practice finding \(a\) and \(d\) from different lists of numbers. You've got this!