# Geometric sequences and infinite geometric series

> IB Mathematics AA SL · Number & Algebra
> Source: https://www.owlsprep.com/study/ib-math-aa-sl-u1-geometric-sequences-and-infinite-geometric/

This sub-topic covers identifying geometric sequences, calculating their nth term, finding sums of finite and infinite geometric series, and testing convergence. These concepts are regularly tested in both IB AA SL Papers 1 and 2.

**Prerequisites:** [Arithmetic sequences and series](https://www.owlsprep.com/study/ib-math-aa-sl-u1-arithmetic-sequences-and-series/); [Exponent rules](https://www.owlsprep.com/study/ib-math-aa-sl-u1-exponents-and-logarithms/)

## Learning objectives

- Identify geometric sequences and calculate the nth term for any given term number
- Calculate the sum of the first n terms of a finite geometric sequence
- Determine if an infinite geometric series converges and calculate its sum if it does
- Apply geometric sequences to real-world problems like compound interest

## Geometric sequences: definition and nth term

**Geometric Sequence** — A sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio $r$.

*Notation:* u_n = r \cdot u_{n-1}

*Example:* 2, 6, 18, 54,... has common ratio $r=3$

By repeatedly multiplying by the common ratio, we can build up the general formula for the nth term of any geometric sequence, starting with first term $u_1$:

$$u_n = u_1 r^{n-1}$$

**Worked example:** The 3rd term of a geometric sequence is 12, and the 5th term is 48. Find all possible values of the common ratio $r$.

1. Write the nth term formula for the 3rd and 5th terms:
2. $$u_3 = u_1 r^2 = 12 \\ u_5 = u_1 r^4 = 48$$
3. Divide the 5th term by the 3rd term to eliminate $u_1$:
4. $$\frac{u_1 r^4}{u_1 r^2} = \frac{48}{12} \implies r^2 = 4$$
5. Solve for $r$. Both positive and negative solutions are valid unless excluded by the problem:
6. $$r = 2 \quad \text{or} \quad r = -2$$

## Sum of a finite geometric series

The sum $S_n$ of the first $n$ terms of a geometric sequence can be written in a closed form, avoiding adding all terms individually. Two different formulas are used depending on whether $r=1$ or not.

**Sum of a Finite Geometric Series** — The total of the first $n$ terms of a geometric sequence, calculated via the following formulas:

*Example:* Sum of first 4 terms of 2, 6, 18,... is 2+6+18+54 = 80

$$\text{If } r \neq 1: \quad S_n = \frac{u_1(1 - r^n)}{1 - r} = \frac{u_1(r^n - 1)}{r - 1} \\ \text{If } r = 1: \quad S_n = n u_1$$

> **tip**
>
> Use the first form $(1-r^n)/(1-r)$ when $|r| < 1$ to avoid working with negative numbers in your calculation.

**Worked example:** Find the sum of the first 10 terms of the sequence 3, 6, 12, 24, ...

1. Identify the values of $u_1$, $r$, and $n$:
2. $$u_1 = 3, \quad r = \frac{6}{3} = 2, \quad n = 10$$
3. Substitute into the sum formula for $r > 1$:
4. $$S_{10} = \frac{3(2^{10} - 1)}{2 - 1}$$
5. Calculate the result:
6. $$S_{10} = 3(1024 - 1) = 3 \times 1023 = 3069$$

## Infinite geometric series and convergence

An infinite geometric series is the sum of all terms of an infinite geometric sequence. Only convergent infinite series have a finite sum; divergent series grow without bound or oscillate infinitely.

**Convergence Condition** — An infinite geometric series converges (has a finite sum) if and only if the absolute value of the common ratio is less than 1.

*Example:* $r = 1/2$ converges, $r = 2$ diverges, $r = -1/2$ converges

$$\text{Converges if } |r| < 1 \\ \text{If convergent, sum to infinity:} \quad S_\infty = \frac{u_1}{1 - r}$$

> **info**
>
> Convergence depends on the magnitude of $r$, not its sign. A negative $r$ will still converge as long as $|r| < 1$.

**Worked example:** Does the infinite series $9 + 3 + 1 + \frac{1}{3} + ...$ converge? If it converges, find its sum.

1. Calculate the common ratio:
2. $$r = \frac{3}{9} = \frac{1}{3}$$
3. Check the convergence condition:
4. $$|r| = \frac{1}{3} < 1 \implies \text{The series converges}$$
5. Substitute into the infinite sum formula:
6. $$S_\infty = \frac{9}{1 - \frac{1}{3}} = \frac{9}{\frac{2}{3}} = \frac{27}{2} = 13.5$$

## Applications to real-world problems

Geometric sequences are widely used to model repeated percentage change, compound growth, depreciation, and recurring investments. The key is to recognize the constant multiplicative factor each period, which is your common ratio.

**Worked example:** A student invests \$1000 in a savings account that pays 5% annual compound interest, with interest reinvested each year. How much money is in the account after 5 years, with no withdrawals?

1. A 5% annual increase means the total amount is multiplied by 1.05 each year, so this is a geometric sequence with:
2. $$u_0 = 1000 \quad (\text{initial investment}), \quad r = 1.05$$
3. The amount after 5 years is the 5th term of the sequence:
4. $$u_5 = u_0 r^5$$
5. Calculate the final amount:
6. $$u_5 = 1000 \times (1.05)^5 \approx 1000 \times 1.2763 = \$1276.28$$

## Common pitfalls

- **Wrong:** Forgetting that $r$ can be negative, only reporting the positive solution.
  - Why it fails: Most problems that solve for $r$ via a quadratic equation have two valid solutions, unless the problem explicitly states all terms are positive.
  - Correct: Always check if both positive and negative values of $r$ satisfy the problem conditions, and include both if valid.
- **Wrong:** Using the infinite sum formula when $|r| \geq 1$.
  - Why it fails: Students often memorize $S_\infty = \frac{u_1}{1-r}$ and use it regardless of the convergence condition.
  - Correct: Always check $|r| < 1$ first, and explicitly state that the infinite sum does not exist if the condition fails.
- **Wrong:** Using $r^n$ instead of $r^{n-1}$ in the nth term formula.
  - Why it fails: Counting term numbers incorrectly leads to off-by-one errors.
  - Correct: Verify the formula with $n=1$: $u_1 = u_1 r^{1-1} = u_1 r^0 = u_1$, which confirms the exponent is $n-1$.
- **Wrong:** Confusing finite and infinite sum formulas, writing $S_\infty = \frac{u_1(1-r^n)}{1-r}$.
  - Why it fails: When $|r| < 1$, $r^n$ approaches 0 as $n$ goes to infinity, so the exponent term disappears.
  - Correct: Remember that the sum to infinity formula simplifies to $\frac{u_1}{1-r}$, with no $n$ or $r^n$ term.

## Cheatsheet

| Concept | Formula | Key Condition |
| --- | --- | --- |
| Geometric sequence nth term | $u_n = u_1 r^{n-1}$ | $r$ = common ratio |
| Sum of $n$ finite terms ($r \neq 1$) | $S_n = \frac{u_1(1-r^n)}{1-r}$ | Any $r \neq 1$ |
| Sum of $n$ finite terms ($r = 1$) | $S_n = n u_1$ | Only when $r=1$ |
| Infinite series convergence | - | $\|r\| < 1$ |
| Sum to infinity (convergent) | $S_\infty = \frac{u_1}{1-r}$ | Only if $\|r\| < 1$ |

## What's next

Geometric sequences and series are a core foundation for financial mathematics, where they are used to model compound growth, depreciation, and annuities. Convergence of geometric series also introduces the key concept of infinite series limits that underpins future topics like calculus and series expansions. This topic is regularly assessed as a standalone question in IB AA SL, often mixed with arithmetic sequence problems to test your ability to distinguish between the two types. Mastering the convergence condition and the different formulas here will make applied problems much easier on exam day.

- [Arithmetic sequences and series](https://www.owlsprep.com/study/ib-math-aa-sl-u1-arithmetic-sequences-and-series/)
- [Laws of exponents and logarithms](https://www.owlsprep.com/study/ib-math-aa-sl-u1-laws-of-exponents-and-logarithms/)
- [Solving exponential and logarithmic equations](https://www.owlsprep.com/study/ib-math-aa-sl-u1-solving-exponential-and-logarithmic-equations/)

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