# Arithmetic sequences and series

> IB Mathematics: Analysis and Approaches SL · Number & Algebra
> Source: https://www.owlsprep.com/study/ib-math-aa-sl-u1-arithmetic-sequences-and-series/

This sub-topic covers core properties of arithmetic sequences and series, including formulas for the nth term and sum of the first n terms. You will learn to apply these to solve real-world and abstract problems common in IB AA SL exams.

**Prerequisites:** [Basic linear equation solving](https://www.owlsprep.com/study/ib-math-aa-sl-algebra-linear-equations/); Term notation for sequences

## Learning objectives

- Identify arithmetic sequences and calculate the common difference
- Find the nth term of an arithmetic sequence using the core formula
- Calculate the sum of the first n terms of an arithmetic series
- Solve practical and abstract exam problems involving arithmetic sequences

## Introduction to Arithmetic Sequences

**Arithmetic Sequence** — An arithmetic sequence has a constant difference between consecutive terms. This difference is called the common difference, calculated as $d = u_{n+1} - u_n$ for all $n$.

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

*Example:* 2, 5, 8, 11,... is arithmetic with $d=3$

Arithmetic sequences can be increasing (positive $d$), decreasing (negative $d$), or constant ($d=0$, all terms equal). Always confirm the difference is constant across the whole sequence to classify it as arithmetic.

**Worked example:** Determine if the sequence $18, 14, 10, 5,...$ is arithmetic.

1. Calculate the difference between the first two terms:
2. $$u_2 - u_1 = 14 - 18 = -4$$
3. Calculate the difference between the next two terms:
4. $$u_3 - u_2 = 10 - 14 = -4$$
5. Calculate the next difference to confirm:
6. $$u_4 - u_3 = 5 - 10 = -5$$
7. Since the difference is not constant across all consecutive terms:
8. The sequence is **not** arithmetic.

> **Exam tip:** Always check at least two pairs of consecutive terms to confirm a sequence is arithmetic.

## The nth Term of an Arithmetic Sequence

**nth Term Formula** — The nth term of any arithmetic sequence is given by the formula below, derived by adding the common difference $(n-1)$ times to the first term:

*Example:* $u_n = u_1 + (n-1)d$

This formula can be rearranged to find any unknown value ($u_1$, $d$, or $n$) when the other three are known. This is one of the most commonly tested skills for this sub-topic in IB exams.

**Worked example:** The 4th term of an arithmetic sequence is 12, and the 9th term is 27. Find $u_1$ and $d$.

1. Write equations for both terms using the nth term formula:
2. $$\begin{align} u_4 &= u_1 + 3d = 12 \\ u_9 &= u_1 + 8d = 27 \end{align}$$
3. Subtract the first equation from the second to eliminate $u_1$:
4. $$(u_1 + 8d) - (u_1 + 3d) = 27 - 12 \\ 5d = 15 \\ d = 3$$
5. Substitute $d=3$ back into the first equation to find $u_1$:
6. $$u_1 + 3(3) = 12 \\ u_1 = 12 - 9 = 3$$
7. Final result: $u_1=3$, $d=3$

> **tip**
>
> If you forget whether the formula uses $n$ or $(n-1)$, test it for $n=1$. If you get $u_1 = u_1$, the formula is correct.

## Sum of an Arithmetic Series

**Sum of the First n Terms** — There are two equivalent formulas for the sum of an arithmetic series, used in different scenarios:

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

*Example:* 1. $S_n = \frac{n}{2}(u_1 + u_n)$ (when you know first/last term)\n2. $S_n = \frac{n}{2}(2u_1 + (n-1)d)$ (when you know $u_1$ and $d$)

Arithmetic series are commonly used to model real-world scenarios, such as regular incremental savings, stacked objects, or monthly payments that increase by a constant amount each period.

**Worked example:** Liam saves \$100 in January, \$110 in February, \$120 in March, and so on, increasing his monthly savings by \$10 each month. What is his total savings after 1 year?

1. Identify the known values: 1 year = 12 months, so $n=12$. $u_1 = 100$, $d=10$.
2. Use the second sum formula since we know $u_1$ and $d$:
3. $$S_n = \frac{n}{2}\left(2u_1 + (n-1)d\right)$$
4. Substitute the values and calculate:
5. $$S_{12} = \frac{12}{2}\left(2(100) + 11(10)\right) = 6\left(200 + 110\right) = 6(310) = 1860$$
6. Liam's total savings after 1 year is \$1860.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Using $n$ instead of $(n-1)$ in the nth term or sum formula
  - Why it fails: You only add the common difference $(n-1)$ times to reach the nth term starting from the first term
  - Correct: Always confirm by testing $n=1$; if $u_1$ matches, the formula is set up correctly
- **Wrong:** Calculating $d$ as positive when the sequence is decreasing
  - Why it fails: Students often subtract the previous term from the next term in reverse order
  - Correct: Always calculate $d = u_{n+1} - u_n$ (next term minus previous term) to get the correct sign
- **Wrong:** Miscounting the number of terms $n$ in word problems
  - Why it fails: For example, counting from month 1 to month 12 as 11 terms instead of 12
  - Correct: Always explicitly count the first term as $n=1$, and double-check the total number of terms before calculating
- **Wrong:** Mixing up arithmetic and geometric sequence formulas
  - Why it fails: Confusing constant addition (arithmetic) with constant multiplication (geometric)
  - Correct: Check if the difference between terms is constant (arithmetic) or the ratio is constant (geometric) before applying a formula

## Cheatsheet

| Concept | Formula | Best For |
| --- | --- | --- |
| Common difference | d = u_{n+1} - u_n | Checking if a sequence is arithmetic |
| nth term | u_n = u_1 + (n-1)d | Finding any term in the sequence |
| Sum (know u₁ + uₙ) | S_n = \frac{n}{2}(u_1 + u_n) | When first and last term are given |
| Sum (know u₁ + d) | S_n = \frac{n}{2}(2u_1 + (n-1)d) | When first term and difference are given |

## What's next

Arithmetic sequences and series are the foundation for all sequence and series work in IB AA SL. The structure you learn here (identifying sequence types, finding terms, calculating sums) is very similar to geometric sequences, the next core sequence type you will study. Both are often tested together in exam questions, and you will use these concepts when learning sigma notation for writing sums of sequences concisely. Mastery of this sub-topic is essential for all further work on sequences, which makes up a significant portion of the Number & Algebra unit.

- [Geometric sequences and infinite geometric series](https://www.owlsprep.com/study/ib-math-aa-sl-u1-geometric-sequences-and-infinite-geometric/)
- [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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