Geometric sequences, series and infinite sums

Introduction to Geometric Sequences and Series

Welcome to the world of geometric growth! While arithmetic sequences (which you might have studied already) involve adding the same number over and over, geometric sequences are all about multiplication. Think about how a viral video spreads, how bacteria multiply, or how a bouncy ball loses height with each bounce—these are all real-world examples of geometric patterns.

In this chapter, we will learn how to identify these patterns, calculate specific terms, add them up, and even explore the "mathematical magic" of adding an infinite number of terms to get a finite result.

1. What is a Geometric Sequence?

A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a fixed, non-zero constant called the common ratio, denoted by the letter \(r\).

For example, in the sequence \(3, 6, 12, 24, ...\), we are multiplying by \(2\) each time. Therefore, the common ratio \(r = 2\).

How to find the Common Ratio (\(r\))

If you aren't sure what the ratio is, simply divide any term by the term that comes immediately before it:

\(r = \frac{u_2}{u_1} = \frac{u_3}{u_2} = \frac{u_{n}}{u_{n-1}}\)

Example: In the sequence \(10, -5, 2.5, -1.25, ...\)
\(r = \frac{-5}{10} = -0.5\)
(Yes, the common ratio can be negative or a fraction!)

The General Term Formula (\(u_n\))

To find any specific term (the \(n^{th}\) term) without listing them all out, we use the formula:

\(u_n = u_1 r^{n-1}\)

Where:
\(u_n\) is the value of the \(n^{th}\) term.
\(u_1\) is the very first term.
\(r\) is the common ratio.
\(n\) is the position of the term (e.g., for the 10th term, \(n=10\)).

Common Mistake: Don't forget the power is \(n-1\), not \(n\). This is because to get to the 5th term, you only need to multiply by the ratio 4 times starting from the 1st term.

Key Takeaway

A geometric sequence grows (or shrinks) by a multiplier. Identify \(u_1\) and \(r\), and you can find any term in the sequence.

2. Geometric Series (Finite Sums)

When we add the terms of a geometric sequence together, it becomes a geometric series. The sum of the first \(n\) terms is denoted as \(S_n\).

The Sum Formula

The IB formula booklet provides two versions of this formula. They are mathematically identical, but one is often easier to use depending on the value of \(r\):

\(S_n = \frac{u_1(r^n - 1)}{r - 1}\) (Usually easier if \(r > 1\))
\(S_n = \frac{u_1(1 - r^n)}{1 - r}\) (Usually easier if \(r < 1\))

Step-by-Step Example: Find the sum of the first 6 terms of \(2, 6, 18, ...\)
1. Identify \(u_1 = 2\).
2. Identify \(r = \frac{6}{2} = 3\).
3. We want the sum of 6 terms, so \(n = 6\).
4. Plug into the formula: \(S_6 = \frac{2(3^6 - 1)}{3 - 1} = \frac{2(729 - 1)}{2} = 728\).

Sigma (\(\Sigma\)) Notation

You might see a series written using the Greek letter Sigma (\(\Sigma\)). This is just a compact way of saying "add these up."

\(\sum_{k=1}^{n} u_1 r^{k-1}\)

This means: "Start with \(k=1\), plug it into the formula, then do \(k=2\), \(k=3\), all the way up to \(n\), and add all the results together."

Key Takeaway

Use the \(S_n\) formula to find the total of a specific number of terms. If you have a calculator (Paper 2), you can often use the \(\Sigma\) function directly!

3. Infinite Geometric Series (Convergent)

What happens if we keep adding terms forever? Usually, the sum would become infinitely large. However, if the terms get smaller and smaller fast enough, the sum actually approaches a specific number. We call this convergence.

Condition for Convergence

An infinite geometric series only has a finite sum if the absolute value of the common ratio is less than 1:

\(|r| < 1\) (which means \(-1 < r < 1\))

If \(r = 2\), the sequence \(2, 4, 8, 16...\) gets bigger and bigger (diverges).
If \(r = 0.5\), the sequence \(10, 5, 2.5, 1.25...\) gets smaller and smaller (converges).

The Sum to Infinity Formula (\(S_\infty\))

If the condition \(|r| < 1\) is met, the sum of all terms forever is:

\(S_\infty = \frac{u_1}{1 - r}\)

Did you know? This is why a bouncing ball eventually "stops." Even though it theoretically bounces an infinite number of times, the total distance it travels is a finite number!

Key Takeaway

Before calculating an infinite sum, always check if \(|r| < 1\). If it isn't, the sum is simply "undefined" or "infinite."

4. Problem Solving and Applications

In your exams, you might not always be given \(u_1\) and \(r\) directly. You might have to solve a "puzzle."

Finding \(u_1\) and \(r\) from two terms

If you know the 3rd term is 20 and the 6th term is 160:
1. Write two equations: \(u_3 = u_1 r^2 = 20\) and \(u_6 = u_1 r^5 = 160\).
2. Divide them: \(\frac{u_1 r^5}{u_1 r^2} = \frac{160}{20}\).
3. Simplify: \(r^3 = 8\), which means \(r = 2\).
4. Substitute \(r\) back into an equation to find \(u_1\): \(u_1(2^2) = 20 \implies 4u_1 = 20 \implies u_1 = 5\).

Real-World Contexts

Geometric sequences often appear in:

• Population growth: If a population grows by 5% each year, \(r = 1.05\).
• Depreciation: If a car loses 10% value each year, \(r = 0.90\).
• Finance: Compound interest (though this is covered in detail in the "Financial Applications" chapter, the math is geometric!).

Quick Review Box

Common Ratio: \(r = \frac{u_{n+1}}{u_n}\)
n-th Term: \(u_n = u_1 r^{n-1}\)
Finite Sum: \(S_n = \frac{u_1(r^n - 1)}{r - 1}\)
Infinite Sum: \(S_\infty = \frac{u_1}{1 - r}\) (only if \(|r| < 1\))

Summary of Common Pitfalls

1. Mixing up Arithmetic and Geometric: Always check if you are adding a constant or multiplying by a constant first.
2. The \(n-1\) Error: In the \(u_n\) formula, the exponent is \(n-1\). In the \(S_n\) formula, the exponent for \(r\) is \(n\).
3. Negative Ratios: If \(r\) is negative, the terms will alternate between positive and negative (e.g., \(2, -4, 8, -16\)). Don't let the negative signs confuse your calculations!
4. Calculator Settings: When using a GDC for Sigma notation, ensure you've typed the formula exactly as written, including the brackets.