Welcome to Geometric Sequences!
In this chapter, we are moving from simple addition-based patterns into the world of Geometric Sequences (also known as a Geometric Progression or GP). Instead of adding the same number every time, we will be multiplying by a constant factor. These sequences are everywhere—from how bacteria grow to how compound interest builds up in a savings account. Don’t worry if the formulas look a bit intimidating at first; we’ll break them down step-by-step!
1. What is a Geometric Sequence?
A Geometric Sequence is a pattern of numbers where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
Key Terms:
- First term \( (a) \): The number the sequence starts with.
- Common ratio \( (r) \): The number we multiply by to get to the next term.
Example: 3, 6, 12, 24, 48...
In this sequence, the first term \( a = 3 \). To get from 3 to 6, or 6 to 12, we multiply by 2. So, the common ratio \( r = 2 \).
Finding the n-th Term
If you want to find any specific term (the n-th term) without writing out the whole list, we use this formula:
\( u_n = ar^{n-1} \)
Step-by-step: Why is it \( n-1 \)?
1. The 1st term is just \( a \).
2. The 2nd term is \( a \times r \).
3. The 3rd term is \( a \times r \times r \), which is \( ar^2 \).
Notice that for the 3rd term, we only multiplied by \( r \) twice. That is why we always use one less than the term number!
Quick Review: To find \( r \), just divide any term by the term before it: \( r = \frac{u_2}{u_1} \).
2. Geometric Series (The Sum)
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 \( S_n \).
There are two versions of the formula for \( S_n \). They are actually the same, but using the right one makes your calculations easier:
1. Use \( S_n = \frac{a(1-r^n)}{1-r} \) when \( r < 1 \).
2. Use \( S_n = \frac{a(r^n-1)}{r-1} \) when \( r > 1 \).
Pro-Tip: Using the version that keeps the denominator positive prevents "negative number headaches" during your exam!
Key Takeaway: A sequence is a list of numbers; a series is those numbers added together.
3. Convergence and the Sum to Infinity
Some geometric sequences keep getting bigger and bigger (divergent), but others get smaller and smaller, heading towards zero (convergent).
When does a series converge?
A geometric series only "settles down" and converges if the common ratio \( r \) is a fraction between -1 and 1. We write this using modulus notation as:
\( |r| < 1 \) (which means \( -1 < r < 1 \)).
Analogy: Imagine you are standing 2 meters from a wall. Each second, you walk half the remaining distance. You get closer and closer, but you’ll never actually go "past" the wall. The total distance you travel is a convergent series.
The Sum to Infinity \( (S_\infty) \)
If a series is convergent, we can calculate what all the infinite terms would add up to using this surprisingly simple formula:
\( S_\infty = \frac{a}{1-r} \)
Common Mistake: Students often try to find \( S_\infty \) for a sequence like 2, 4, 8, 16... You can't! Because \( r=2 \), the numbers keep growing. The formula only works if \( |r| < 1 \).
Did you know? This concept helps explain Zeno's Paradox, an ancient Greek puzzle about how an arrow can ever reach its target if it always has to travel half the remaining distance first!
4. Modelling and Real-World Maths
The OCR syllabus expects you to use these formulas to solve real-world problems, especially those involving money or growth.
Compound Interest
When money grows by a percentage, it is a geometric sequence.
- If an investment grows by 5% each year, the common ratio \( r \) is 1.05.
- If a car loses 10% of its value each year, the common ratio \( r \) is 0.90.
Solving for n using Logarithms
Sometimes you’ll be asked: "How many years until the investment exceeds £10,000?" This means you need to solve an inequality where \( n \) is the power.
Step-by-step process:
1. Set up your inequality: \( ar^{n-1} > 10,000 \).
2. Divide by \( a \).
3. Use logarithms to bring the \( n \) down: \( (n-1)\log(r) > \log(\text{something}) \).
4. Solve for \( n \).
Watch out! If you divide an inequality by a negative number (or the log of a number less than 1), you must flip the inequality sign.
Final Quick Review Box
- n-th term: \( u_n = ar^{n-1} \)
- Sum of \( n \) terms: \( S_n = \frac{a(1-r^n)}{1-r} \)
- Sum to infinity: \( S_\infty = \frac{a}{1-r} \) (Only if \( |r| < 1 \))
- Geometric growth: \( r = 1 + \text{percentage} \)
- Geometric decay: \( r = 1 - \text{percentage} \)
Don't worry if this seems tricky at first! The most important skill is identifying \( a \) and \( r \) from the question. Once you have those, the formulas do the heavy lifting for you.