Geometric series

Introduction: Welcome to Geometric Series!

Welcome to one of the most practical and fascinating parts of A Level Mathematics. If you’ve ever watched a viral video's views double every hour, or seen how interest builds up in a savings account, you have already encountered geometric growth.

In this chapter, we move away from adding the same number each time (like in Arithmetic series) and look at what happens when we multiply by the same number over and over again. Don't worry if you find the formulas a bit intimidating at first; we will break them down step-by-step until they feel like second nature!

1. What is a Geometric Sequence?

Before we can sum a series, we need to understand the sequence it comes from. A geometric sequence (also called a Geometric Progression or GP) is a list of numbers where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Key Terms to Learn:

• First term (\(a\)): This is simply the number the sequence starts with.
• Common ratio (\(r\)): This is the number we multiply by to get to the next term. You can find it by dividing any term by the one before it: \(r = \frac{u_2}{u_1}\).
• \(n\)th term (\(u_n\)): The specific term at position \(n\).

Example: 3, 6, 12, 24...
Here, the first term \(a = 3\).
The common ratio \(r = 2\) (because \(3 \times 2 = 6\), \(6 \times 2 = 12\), and so on).

The Formula for the \(n\)th Term

To find any term in the sequence without listing them all, we use:
\(u_n = ar^{n-1}\)

Why \(n-1\)? Think of it this way: to get to the 2nd term, you multiply by \(r\) once. To get to the 3rd term, you multiply by \(r\) twice. So, to get to the \(n\)th term, you always multiply by \(r\) one fewer time than the position number!

Quick Review:

Sequence: 5, -10, 20, -40...
\(a = 5\)
\(r = -2\) (Yes, \(r\) can be negative! This makes the signs alternate).
5th term: \(u_5 = 5 \times (-2)^{4} = 5 \times 16 = 80\).

Key Takeaway: A geometric sequence is all about multiplication. Always find \(a\) and \(r\) first; they are the "keys" to unlocking any problem.

2. Geometric Series: Finding the Sum

A series is what you get when you add up the terms of a sequence. For example, if the sequence is 2, 4, 8, the series is \(2 + 4 + 8 = 14\). We use the notation \(S_n\) to represent the sum of the first \(n\) terms.

The Sum Formula

There are two ways to write the same formula. You can use whichever you find easier, but usually:

• Use this if \(r > 1\) (to keep numbers positive): \(S_n = \frac{a(r^n - 1)}{r - 1}\)
• Use this if \(r < 1\): \(S_n = \frac{a(1 - r^n)}{1 - r}\)

Don't worry if this seems tricky... The most common mistake is confusing the \(n\)th term formula (\(r^{n-1}\)) with the Sum formula (\(r^n\)). Just remember: "Terms need a head-start (\(n-1\)), but Sums use the whole amount (\(n\))."

Step-by-Step Example:

Find the sum of the first 8 terms of: 2, 6, 18, 54...
1. Identify \(a = 2\).
2. Identify \(r = 6 \div 2 = 3\).
3. We want \(n = 8\).
4. Plug into the formula: \(S_8 = \frac{2(3^8 - 1)}{3 - 1}\)
5. Simplify: \(S_8 = \frac{2(6561 - 1)}{2} = 6560\).

Key Takeaway: The sum formula allows us to add up huge sequences instantly. Just be careful with your calculator brackets!

3. Convergent Series and Sum to Infinity

Imagine you are standing 2 meters from a wall.
First, you walk 1 meter (halfway).
Then, you walk 0.5 meters (half of what's left).
Then, you walk 0.25 meters...

Will you ever actually punch through the wall? No! Your total distance will get closer and closer to 2 meters, but never exceed it. This is a convergent series.

When does a series converge?

A geometric series only has a Sum to Infinity (\(S_\infty\)) if the common ratio \(r\) is between -1 and 1.
In math notation, we write this as: \(|r| < 1\).

If \(r = 2\), the numbers just keep getting bigger and bigger (divergent), so the sum would be infinity!

The Sum to Infinity Formula

If \(|r| < 1\), the sum is incredibly simple:
\(S_\infty = \frac{a}{1 - r}\)

Did you know?

This formula is why recurring decimals work! The decimal \(0.3333...\) is actually a geometric series: \(\frac{3}{10} + \frac{3}{100} + \frac{3}{1000}...\) where \(a = 0.3\) and \(r = 0.1\). Using the formula: \(S_\infty = \frac{0.3}{1 - 0.1} = \frac{0.3}{0.9} = \frac{1}{3}\).

Key Takeaway: You can only find \(S_\infty\) if the numbers are getting smaller (specifically, if \(|r| < 1\)).

4. Common Pitfalls and Tips

• The "n-1" Trap: Remember \(u_n\) uses \(r^{n-1}\), but \(S_n\) uses \(r^n\).
• Negative Ratios: If \(r\) is negative, like -0.5, your terms will jump between positive and negative. When using the sum formula, always put negative numbers in brackets on your calculator: \((-0.5)^n\).
• Solving for \(n\): If a question asks "How many terms are needed for the sum to exceed 1000?", you will likely need to use Logarithms to solve for the power \(n\).

5. Modelling with Geometric Series

In your exam, you might see "real-world" questions. These are usually about money or growth.

Analogy: Compound Interest
If you put £1000 in a bank with 5% interest, next year you have \(1000 \times 1.05\). The year after, you have \((1000 \times 1.05) \times 1.05\).
This is a geometric sequence where \(a = 1000\) and \(r = 1.05\).

Important Tip for Modelling: Read carefully to see if the first term starts at Year 0 or Year 1. This affects whether your power is \(n\) or \(n-1\). Drawing a quick timeline for the first 3 terms often clears up the confusion!

Chapter Summary

• Geometric Sequence: Multiply by \(r\) every time.
• \(n\)th term: \(u_n = ar^{n-1}\).
• Sum of \(n\) terms: \(S_n = \frac{a(1-r^n)}{1-r}\).
• Sum to infinity: \(S_\infty = \frac{a}{1-r}\), provided that \(|r| < 1\).
• Modelling: Use these tools to solve problems involving percentages, population, and finance.