# Arithmetic sequences and series

> IB Mathematics: Applications and Interpretation SL · Number and Algebra
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u1-arithmetic-sequences-and-series/

This module covers core properties of arithmetic sequences and series for IB AI SL. You will learn to identify sequences, calculate individual terms, find partial sums, and solve common real-world problems tested in your exam.

**Prerequisites:** Basic linear equations and algebraic manipulation; Sequence and nth term notation

## Learning objectives

- Identify whether a sequence is arithmetic and find its key parameters
- Calculate the nth term of any arithmetic sequence
- Find the sum of the first n terms of an arithmetic series
- Solve real-world application problems common in IB exams

## What is an Arithmetic Sequence?

**Arithmetic Sequence** — A sequence where the difference between every pair of consecutive terms is constant. This constant is called the common difference.

*Notation:* $u_n$ = nth term, $u_1$ = first term, $d$ = common difference

*Example:* 3, 7, 11, 15... has $u_1=3$, $d=4$

To confirm a sequence is arithmetic, you just calculate the difference between each pair of consecutive terms. If all differences are equal, the sequence is arithmetic. If any difference differs, the sequence is not arithmetic.

**Worked example:** Is 2, 5, 8, 11, 15 an arithmetic sequence?

1. Calculate the difference between the second and first term:

   $$5 - 2 = 3$$
2. Calculate the difference between the third and second term:

   $$8 - 5 = 3$$
3. Calculate the difference between the fourth and third term:

   $$11 - 8 = 3$$
4. Calculate the difference between the fifth and fourth term:

   $$15 - 11 = 4$$
5. The differences are not all equal, so the sequence is not arithmetic.

> **Exam tip:** Always check at least three consecutive difference values to confirm a sequence is arithmetic.

## The General nth Term Formula

We can derive a general formula for the nth term by starting from the definition: the first term is $u_1$, the second term is $u_1 + d$, the third term is $u_1 + 2d$, and so on. To get the nth term, we add $d$ a total of $(n-1)$ times to the first term.

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

> **info**
>
> $d$ can be negative, which means the sequence is decreasing. Always calculate $d$ as second term minus first term to get the correct sign automatically.

**Worked example:** Find the 20th term of the arithmetic sequence 5, 9, 13, 17...

1. Identify the required parameters:

   $$u_1 = 5, \quad d = 9 - 5 = 4, \quad n = 20$$
2. Substitute into the nth term formula:

   $$u_20 = 5 + (20 - 1)(4)$$
3. Simplify to get the final result:

   $$u_20 = 5 + 19 \times 4 = 5 + 76 = 81$$
4. Final answer: The 20th term is 81.

**Check your understanding**

Test your understanding

1. What is the 12th term of the sequence 100, 92, 84...?

   - 12
   - 8
   - -8
   - -12

   *Why:* Correct: $u_1=100$, $d=-8$, so $u_12 = 100 + (11)(-8) = 12$

## Sum of an Arithmetic Series

**Arithmetic Series** — The sum of the first n terms of an arithmetic sequence, denoted $S_n$

*Notation:* S_n

The sum formula comes from pairing the first and last term: every pair adds up to the same value of $u_1 + u_n$, and there are $\frac{n}{2}$ such pairs. There are two equivalent forms of the formula.

$$S_n = \frac{n}{2} (u_1 + u_n)$$

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

**Worked example:** Find the sum of the first 30 terms of the sequence 2, 6, 10, 14...

1. Identify parameters:

   $$u_1 = 2, \quad d = 4, \quad n = 30$$
2. Use the formula with $u_1$ and $d$, since we do not know $u_30$:

   $$S_30 = \frac{30}{2} \left[ 2(2) + (30 - 1)(4) \right]$$
3. Simplify step by step:

   $$S_30 = 15 \left[ 4 + 116 \right] = 15 \times 120 = 1800$$
4. Final answer: The sum of the first 30 terms is 1800.

> **Exam tip:** If you know both the first and last term, use the first sum formula for faster calculation in exams.

## Real-World Applications

IB AI SL exams regularly test arithmetic sequences in real contexts, including fixed monthly savings increments, simple interest, and constant annual depreciation of assets. The key skill is translating the word problem into sequence parameters correctly.

**Worked example:** A person saves \$100 in the first month, and increases their savings by \$20 each month. What is their total savings after 12 months?

1. Translate to sequence parameters: first month savings is $u_1$, monthly increment is $d$, number of months is $n$:

   $$u_1 = 100, \quad d = 20, \quad n = 12$$
2. Total savings is the sum of the first 12 terms, so substitute into the sum formula:

   $$S_12 = \frac{12}{2} \left[ 2(100) + (12 - 1)(20) \right]$$
3. Simplify:

   $$S_12 = 6 \left[ 200 + 220 \right] = 6 \times 420 = 2520$$
4. Final answer: Total savings after 12 months is \$2520.

> **tip**
>
> Always double-check the value of $n$: if asked for the first k periods, $n = k$, not $k-1$.

## Common pitfalls

- **Wrong:** Using $n$ instead of $n-1$ in the nth term and sum formulas
  - Why it fails: There are only $(n-1)$ common differences between the first term and the nth term, so using $n$ gives an incorrect result
  - Correct: Always write $(n-1)$ in both formulas, double-check this before substituting values
- **Wrong:** Using a positive $d$ for decreasing sequences
  - Why it fails: If the sequence decreases, the difference between consecutive terms is negative
  - Correct: Always calculate $d$ as $u_2 - u_1$ to automatically get the correct sign
- **Wrong:** Miscalculating $n$ when summing from term m to term k
  - Why it fails: Counting $k - m$ terms skips the starting term m
  - Correct: For inclusive counting, use $n = k - m + 1$ to get the correct number of terms
- **Wrong:** Confusing the nth term formula with the sum formula
  - Why it fails: Exam questions often ask for either a single term or a total sum, mixing up formulas leads to losing easy marks
  - Correct: Underline what the question asks for: "find the 10th term" = nth term formula, "find the total of 10 terms" = sum formula
- **Wrong:** Assuming a sequence is arithmetic without checking
  - Why it fails: Exam questions sometimes give geometric sequences to test if you check for constant difference
  - Correct: Always check the difference between at least two pairs of consecutive terms before proceeding

## Cheatsheet

| Concept | Formula | Key Notes |
| --- | --- | --- |
| nth term | $u_n = u_1 + (n-1)d$ | $d = u_2 - u_1$ |
| Sum (know $u_1, u_n$) | $S_n = \frac{n}{2}(u_1 + u_n)$ | Fastest for this case |
| Sum (know $u_1, d$) | $S_n = \frac{n}{2}[2u_1 + (n-1)d]$ | Use when you don't know $u_n$ |
| Number of terms (m to k) | $n = k - m + 1$ | Add 1 for inclusive counting |

## What's next

Arithmetic sequences and series are the foundational discrete sequence type you will use throughout IB AI SL. They share a similar structure to geometric sequences and series, which are the other core sequence type tested on the exam. The logic of fixed constant increments also translates directly to linear growth models, linear regression, and financial applications you will cover later in the course. Mastery of this topic makes learning more advanced sequence and modelling topics much simpler.

- [Geometric sequences and series](https://www.owlsprep.com/study/ib-math-ai-sl-u1-geometric-sequences-and-series/)
- [Financial applications of geometric sequences and series](https://www.owlsprep.com/study/ib-math-ai-sl-u1-financial-applications-of-geometric-sequences/)
- [Laws of exponents with rational exponents](https://www.owlsprep.com/study/ib-math-ai-sl-u1-laws-of-exponents-with-rational/)

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