# Arithmetic and geometric sequences and series

> IB Mathematics AI HL · Number and algebra
> Source: https://www.owlsprep.com/study/ib-math-ai-hl-u1-arithmetic-and-geometric-sequences-and/

This subtopic covers core properties of arithmetic and geometric sequences and series, including nth term formulas, finite and infinite sum calculations, and common application problems for IB AI HL exams.

**Prerequisites:** Basic algebraic manipulation; Exponents and logarithms

## Learning objectives

- Distinguish between arithmetic and geometric sequences and series
- Calculate the nth term for both sequence types
- Find the sum of finite arithmetic and geometric series
- Calculate the sum of convergent infinite geometric series
- Solve IB-style application problems involving sequences

## Arithmetic Sequences

**Arithmetic Sequence** — A sequence where consecutive terms differ by a constant value called the common difference.

*Notation:* $u_n$ = nth term, $d$ = common difference

*Example:* 3, 7, 11, 15, ... has $d = 4$

The general formula for the nth term of an arithmetic sequence is derived from the linear pattern of increasing or decreasing terms:

$$u_n = u_1 + (n-1)d$$

**Worked example:** The 5th term of an arithmetic sequence is 22, and the 10th term is 47. Find the 15th term.

1. Set up simultaneous equations using the nth term formula:
2. $$\begin{align*} u_1 + 4d &= 22 \\ u_1 + 9d &= 47 \end{align*}$$
3. Subtract the first equation from the second to eliminate $u_1$:
4. $$5d = 25 \implies d = 5$$
5. Substitute $d=5$ back to find the first term:
6. $$u_1 + 4(5) = 22 \implies u_1 = 2$$
7. Calculate the 15th term:
8. $$u_{15} = 2 + (15-1)(5) = 72$$

> **Exam tip:** Always remember the number of differences between term 1 and term n is $(n-1)$, not $n$.

*Calculator:* allowed

## Arithmetic Series

**Arithmetic Series** — The sum of the first n terms of an arithmetic sequence.

*Notation:* $S_n$ = sum of first n terms

*Example:* Sum of first 4 terms of 3, 7, 11, 15 is $3+7+11+15 = 36$

The sum formula is derived by pairing the first and last term, which always add to the same constant value. There are two equivalent forms:

$$S_n = \frac{n}{2}\left(u_1 + u_n\right) = \frac{n}{2}\left(2u_1 + (n-1)d\right)$$

**Worked example:** Find the sum of the first 30 terms of the sequence $7, 4, 1, -2,...$

1. Identify values: $u_1 = 7$, $d = -3$, $n = 30$
2. Substitute into the sum formula:
3. $$S_{30} = \frac{30}{2}\left(2(7) + (30-1)(-3)\right)$$
4. Simplify to get the final result:
5. $$15\left(14 - 87\right) = 15(-73) = -1095$$

*Calculator:* allowed

## Geometric Sequences

**Geometric Sequence** — A sequence where each consecutive term is multiplied by a constant common ratio $r$.

*Notation:* $u_n$ = nth term, $r$ = common ratio

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

Repeated multiplication by the common ratio gives the exponential nth term formula for geometric sequences:

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

**Worked example:** A geometric sequence has a third term of 12 and a sixth term of 96. Find the first term.

1. Write equations for both given terms:
2. $$\begin{align*} u_1 r^2 &= 12 \\ u_1 r^5 &= 96 \end{align*}$$
3. Divide the second equation by the first to eliminate $u_1$:
4. $$r^3 = \frac{96}{12} = 8 \implies r = 2$$
5. Substitute $r=2$ back to solve for $u_1$:
6. $$u_1 (2)^2 = 12 \implies 4u_1 = 12 \implies u_1 = 3$$

> **Exam tip:** Common ratio can be negative, leading to alternating positive and negative terms.

*Calculator:* allowed

## Geometric Series (Finite and Infinite)

**Geometric Series** — The sum of the first n terms of a geometric sequence. An infinite sum exists only for convergent series.

*Notation:* $S_n$ = sum of first n terms, $S_\infty$ = infinite sum

The sum of a finite geometric series is given by:

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

An infinite geometric series converges to a finite sum only if the absolute value of the common ratio is less than 1. When this condition is met, the infinite sum is:

$$S_\infty = \frac{u_1}{1 - r}, \quad |r| < 1$$

> **warning**
>
> Never use the infinite sum formula without first checking that $|r| < 1$. If $|r| \geq 1$, the terms grow without bound and no finite sum exists.

**Worked example:** Find the sum to infinity of the series $18 + 6 + 2 + \frac{2}{3} + ...$

1. Identify $u_1 = 18$, calculate common ratio $r = \frac{6}{18} = \frac{1}{3}$
2. Check convergence: $|\frac{1}{3}| < 1$, so infinite sum exists
3. Substitute into the infinite sum formula:
4. $$S_\infty = \frac{18}{1 - \frac{1}{3}} = \frac{18}{\frac{2}{3}} = 27$$

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Using $n$ instead of $n-1$ in nth term formulas
  - Why it fails: The first term occurs at n=1, so the exponent or difference is always one less than the term number
  - Correct: Always write $(n-1)$ for the exponent in geometric nth terms and for the difference in arithmetic nth terms
- **Wrong:** Skipping the convergence check for infinite geometric series
  - Why it fails: The infinite sum formula only works for $|r| < 1$, you will lose marks for not confirming convergence
  - Correct: Always state that $|r| < 1$ before using the infinite sum formula
- **Wrong:** Mixing up arithmetic and geometric sum formulas
  - Why it fails: The formulas are structurally different, and using the wrong one leads to zero marks for the question
  - Correct: Remember: arithmetic is linear (uses $d$ for difference), geometric is exponential (uses $r$ for ratio)
- **Wrong:** Sign errors with negative common ratios
  - Why it fails: Raising a negative number to an even or odd power changes the sign of $r^n$, leading to incorrect sum calculations
  - Correct: Evaluate $r^n$ separately, check the sign before substituting into the sum formula
- **Wrong:** Counting the wrong number of terms between two given terms
  - Why it fails: Some students add an extra 1 when calculating the difference in term positions
  - Correct: The difference in exponents/positions is always $n_2 - n_1$ for two given terms

## Cheatsheet

| Property | Arithmetic | Geometric |
| --- | --- | --- |
| nth term | $u_n = u_1 + (n-1)d$ | $u_n = u_1 r^{n-1}$ |
| Common difference/ratio | $d = u_{n+1} - u_n$ | $r = \frac{u_{n+1}}{u_n}$ |
| Sum of first $n$ terms | $S_n = \frac{n}{2}(2u_1 + (n-1)d)$ | $S_n = \frac{u_1(1-r^n)}{1-r}, r \neq 1$ |
| Infinite sum | No finite sum (unless $d=0$) | $S_\infty = \frac{u_1}{1-r}, \|r\|<1$ |

## What's next

This subtopic is a core foundation for many key topics in IB AI HL, including financial mathematics (compound interest is a geometric sequence application), exponential growth and decay models, and recurrence relation modelling. Sequences and series also frequently combine with logarithms, as solving for the number of terms $n$ in a geometric sequence requires logarithmic manipulation. Mastering these formulas and problem-solving patterns will prepare you for the modelling-focused questions that make up a large portion of IB AI HL exams.

- [Exponential and Logarithmic Functions](https://www.owlsprep.com/study/ib-math-ai-hl-u2-exponential-and-logarithmic-functions/)
- [Financial applications of geometric sequences and series](https://www.owlsprep.com/study/ib-math-ai-hl-u1-financial-applications-of-geometric-sequences/)
- [Laws of Logarithms](https://www.owlsprep.com/study/ib-math-ai-hl-u1-laws-of-logarithms/)

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