Geometric Sequences and Series 📈
Welcome, students! In this lesson, you will learn how geometric sequences and geometric series work, why they appear so often in mathematics and real life, and how to use them in IB Mathematics: Analysis and Approaches HL. A geometric sequence is a list of numbers where each term is found by multiplying the previous term by the same number. A geometric series is the sum of the terms in that sequence. These ideas are useful in finance, population growth, depreciation, and many exam questions. By the end of this lesson, you should be able to identify geometric patterns, write formulas, calculate sums, and explain why the formulas are true using algebraic reasoning.
What is a geometric sequence?
A sequence is an ordered list of numbers. In a geometric sequence, the ratio between consecutive terms is constant. This fixed number is called the common ratio and is usually written as $r$. If the first term is $u_1$, then the next terms are found by multiplying by $r$ again and again:
$$u_1,\; u_1r,\; u_1r^2,\; u_1r^3,\; \dots$$
For example, if the first term is $3$ and the common ratio is $2$, the sequence is
$$3,\; 6,\; 12,\; 24,\; 48,\; \dots$$
Each term is twice the previous one. If the first term is $80$ and the common ratio is $\frac{1}{2}$, the sequence is
$$80,\; 40,\; 20,\; 10,\; 5,\; \dots$$
This type of sequence models repeated growth or repeated shrinkage. For example, a bacterial population might double each hour, while the value of some equipment might drop by the same percentage every year 📉.
The general term, or $n$th term, of a geometric sequence is
$$u_n = u_1r^{n-1}.$$
This formula is important because it lets you jump straight to any term without listing all the terms before it. If $u_1=5$ and $r=3$, then
$$u_4 = 5\cdot 3^{3} = 135.$$
Notice the exponent is $n-1$, not $n$. That is because the first term already uses $r^0$, and $r^0=1$.
How to recognize and use the common ratio
To check whether a sequence is geometric, divide each term by the one before it. If the result is the same every time, the sequence is geometric. For example, look at
$$7,\; -14,\; 28,\; -56,\; \dots$$
The ratios are
$$\frac{-14}{7}=-2,\quad \frac{28}{-14}=-2,\quad \frac{-56}{28}=-2.$$
So the common ratio is $r=-2$. A negative ratio means the terms alternate in sign. That is why the sequence goes positive, negative, positive, negative. This can happen in algebra and in alternating real-world patterns, such as a quantity that flips direction at each step.
Sometimes the common ratio is a fraction. For instance, in the sequence
$$100,\; 75,\; 56.25,\; \dots$$
the ratio is
$$r=\frac{75}{100}=\frac{3}{4}.$$
A ratio between $-1$ and $1$ in absolute value means the terms get smaller in size. If $0<r<1$, the terms stay positive and shrink. If $-1<r<0$, the terms shrink and alternate signs. If $|r|>1$, the terms grow in magnitude.
Be careful not to confuse arithmetic and geometric sequences. In an arithmetic sequence, the difference is constant. In a geometric sequence, the ratio is constant. For example,
$$2,\; 5,\; 8,\; 11,\; \dots$$
is arithmetic because the difference is $3$, while
$$2,\; 6,\; 18,\; 54,\; \dots$$
is geometric because the ratio is $3$.
Geometric series and their sums
A series is the sum of terms in a sequence. The geometric series formed from the first $n$ terms of a geometric sequence is
$$S_n = u_1 + u_1r + u_1r^2 + \cdots + u_1r^{n-1}.$$
This sum has a very useful formula when $r\neq 1$:
$$S_n = \frac{u_1(1-r^n)}{1-r}.$$
An equivalent form is
$$S_n = \frac{u_1(r^n-1)}{r-1}.$$
Both are correct. The first form is often easier when $|r|<1$, and the second may look more natural in some problems.
For example, if the sequence is $4, 12, 36, 108, \dots$, then $u_1=4$ and $r=3$. The sum of the first $4$ terms is
$$S_4 = \frac{4(1-3^4)}{1-3} = \frac{4(1-81)}{-2} = 160.$$
You can also check by adding directly:
$$4+12+36+108=160.$$
The formula saves time, especially when $n$ is large.
Why the formula works
IB HL expects more than just using formulas; you should also understand why they are true. A common proof uses algebra. Start with
$$S_n = u_1 + u_1r + u_1r^2 + \cdots + u_1r^{n-1}.$$
Multiply by $r$:
$$rS_n = u_1r + u_1r^2 + \cdots + u_1r^n.$$
Now subtract:
$$S_n - rS_n = u_1 - u_1r^n.$$
Factor both sides:
$$S_n(1-r)=u_1(1-r^n).$$
Then divide by $1-r$ to get
$$S_n = \frac{u_1(1-r^n)}{1-r}.$$
This algebraic method is a classic example of symbolic manipulation in Number and Algebra.
Infinite geometric series
An infinite geometric series has infinitely many terms:
$$u_1 + u_1r + u_1r^2 + u_1r^3 + \cdots$$
Not every infinite geometric series has a sum. It only converges if
$$|r|<1.$$
In that case, the sum to infinity is
$$S_\infty = \frac{u_1}{1-r}.$$
Why does this happen? If $|r|<1$, then $r^n\to 0$ as $n\to\infty$. Starting from the finite sum formula,
$$S_n = \frac{u_1(1-r^n)}{1-r},$$
taking the limit gives
$$S_\infty = \lim_{n\to\infty}S_n = \frac{u_1}{1-r}.$$
For example, consider
$$8 + 4 + 2 + 1 + \cdots$$
Here $u_1=8$ and $r=\frac{1}{2}$. Since $|r|<1$,
$$S_\infty = \frac{8}{1-\frac{1}{2}} = 16.$$
This is surprising at first: infinitely many positive terms add up to a finite number. That idea appears in mathematical models, especially when something gets smaller and smaller each step.
If $|r|\ge 1$, the infinite geometric series does not have a finite sum. For example, $1+2+4+8+\dots$ grows without bound, and $1-1+1-1+\dots$ does not approach one stable total.
Applications and IB-style reasoning
Geometric sequences and series appear in practical situations where change happens by a fixed percentage. A savings account with compound interest is a strong example. If an account grows by $5\%$ per year, then each year the balance is multiplied by $1.05$. If the starting balance is $1000$, then after $n$ years the balance is
$$1000(1.05)^n.$$
This is a geometric sequence with $u_1=1000$ and $r=1.05$. If yearly deposits are made, geometric series can help find the total accumulated value.
Another example is depreciation. If a car loses $20\%$ of its value each year, then the multiplier is $0.8$. If the car is worth $20000$ now, then after $3$ years its value is
$$20000(0.8)^3=10240.$$
In exam questions, you may need to find $u_1$, $r$, or $n$ from information given in words or equations. You may also need to solve equations like
$$u_1r^{n-1}=\text{given value}$$
or
$$\frac{u_1(1-r^n)}{1-r}=\text{given sum}.$$
These often require rearranging with logarithms or careful algebra. For example, if
$$3\cdot 2^{n-1}=96,$$
then
$$2^{n-1}=32=2^5,$$
so
$$n-1=5\quad \Rightarrow \quad n=6.$$
This shows how geometric sequences connect to exponential equations, another important part of Number and Algebra.
Conclusion
Geometric sequences and series are built on repeated multiplication. The key ideas are the common ratio $r$, the general term $u_n=u_1r^{n-1}$, the finite sum formula $S_n=\frac{u_1(1-r^n)}{1-r}$, and the infinite sum formula $S_\infty=\frac{u_1}{1-r}$ when $|r|<1$. students, these tools help you describe growth, decay, and repeated processes in a precise way. They also strengthen your algebraic skill because you must manipulate formulas, solve equations, and justify steps clearly. In IB Mathematics: Analysis and Approaches HL, geometric sequences and series are not just a memorization topic; they are a bridge between patterns, algebra, and real-world modelling 🌟.
Study Notes
- A geometric sequence has a constant ratio between consecutive terms.
- The constant ratio is called the common ratio, written as $r$.
- The $n$th term is $u_n=u_1r^{n-1}$.
- A geometric series is the sum of the terms of a geometric sequence.
- The finite sum formula is $S_n=\frac{u_1(1-r^n)}{1-r}$ for $r\neq 1$.
- An equivalent form is $S_n=\frac{u_1(r^n-1)}{r-1}$.
- The sum to infinity exists only when $|r|<1$.
- If $|r|<1$, then $S_\infty=\frac{u_1}{1-r}$.
- A negative $r$ makes the terms alternate in sign.
- A ratio with $|r|>1$ makes the terms grow in magnitude.
- Geometric sequences model repeated percentage growth or decay.
- In IB HL, you should be able to identify, calculate, and justify results using algebraic reasoning.