# Arithmetic Sequences and Series

> Number patterns that add the same amount each step. The nth term, the sum of n terms and sigma notation.

From HSC Maths Advanced on WhizzNotes, free study notes for NSW students: https://whizznotes.com/mathematics-advanced/arithmetic-sequences/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The savings jar
Isla puts 5 dollars in a jar in week 1. Each week after that she puts in 3 dollars more than the week before: 5, 8, 11, 14... She wonders two things. How much will she put in during week 20? And how much will be in the jar altogether by then?

She could write out all twenty numbers and add them. Or she could use two short formulas. Patterns where you add the same amount each time are called **arithmetic sequences**, and they appear in pay rises, stacked seats in a stadium and savings plans.

**In short:** An arithmetic sequence adds the same amount $d$ each time, like 3, 7, 11, 15. Its $n$th term is $T_n = a + (n - 1)d$, and the sum of its first $n$ terms is $S_n = \dfrac{n}{2}[2a + (n - 1)d]$.

## Starting point: Years 9 and 10

### Number patterns

A **sequence** is a list of numbers in order. Each number is a **term**. The first term is $T_1$, the second is $T_2$, and so on.

To continue a pattern, find the rule. In 4, 7, 10, 13, ..., each term is 3 more than the last.

### Calculator tip: List terms with Ans
Type the first term and press `=`. Then type `+ 4` (the common difference) and press `=` again and again. Each press adds 4 to the previous answer, so the terms appear one by one. This is a quick way to check a formula for $T_n$.

### Finding a rule for the nth term

If the difference is always 3, the rule has $3n$ in it. Compare:

| $n$ | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| $3n$ | 3 | 6 | 9 | 12 |
| term | 4 | 7 | 10 | 13 |

Each term is one more than $3n$, so $T_n = 3n + 1$. Then $T_{50} = 151$.

### Worked example: Does 200 belong?
Is 200 a term of $T_n = 3n + 1$? Solve $3n + 1 = 200$: $n = 66.33$. That is not a whole number, so 200 is not a term.

### Patterns and straight lines

Plot $n$ against $T_n$ and the points lie on a straight line. The common difference is the gradient. That is why arithmetic sequences are sometimes called **linear** sequences.

### Watch out: Term number versus term value
$n$ is the position (1st, 2nd, 3rd). $T_n$ is the value at that position. Keep them separate.

**Mind map**

```text
Arithmetic sequences
  Common difference d
    Same amount added
    d = T₂ - T₁
  General term
    Tₙ = a + (n - 1)d
    Linear pattern
  Series
    Sum of terms
    Sₙ = n/2 (a + l)
    Sₙ = n/2 (2a + (n-1)d)
  Sigma notation
    Σ means add up
```

### Year 11: The general term
An **arithmetic sequence** has first term $a$ and **common difference** $d$:

$$a, \; a + d, \; a + 2d, \; a + 3d, \; \dots$$

To reach the $n$th term you add $d$ a total of $n - 1$ times:

$$T_n = a + (n - 1)d$$

**Isla's week 20:** $a = 5$, $d = 3$, so $T_{20} = 5 + 19 \times 3 = 62$ dollars.

**Finding $a$ and $d$ from two terms.** If $T_4 = 17$ and $T_{10} = 41$, then $6d = 24$, so $d = 4$ and $a = 17 - 12 = 5$.

**Three terms in a row** $x$, $y$, $z$ are arithmetic if $y - x = z - y$, so the middle term is the average: $y = \dfrac{x + z}{2}$.

### Year 12: Arithmetic series
A **series** is the sum of the terms of a sequence. Write the sum forwards and backwards and add: every pair adds to $a + l$, where $l$ is the last term. So

$$S_n = \frac{n}{2}(a + l) = \frac{n}{2}\big[2a + (n - 1)d\big]$$

**Isla's jar after 20 weeks:** $S_{20} = \dfrac{20}{2}(5 + 62) = 670$ dollars.

**Sigma notation.** $\displaystyle\sum_{k=1}^{n} T_k$ means "add $T_k$ for $k = 1$ to $n$."

$$\sum_{k=1}^{4}(2k + 1) = 3 + 5 + 7 + 9 = 24$$

**How many terms?** How many terms of $3 + 7 + 11 + \dots$ add to 300? Solve $\dfrac{n}{2}[6 + 4(n - 1)] = 300$, which is $2n^2 + n - 300 = 0$, giving $n = 12$ (ignore the negative root).

**A term from the sum formula.** If $S_n$ is given, then $T_n = S_n - S_{n-1}$, because the sum of $n$ terms minus the sum of $n - 1$ terms leaves the last term. For $S_n = n^2 + 4n$: $T_{10} = S_{10} - S_9 = 140 - 117 = 23$.

**Inequalities in $n$.** For "the first row with more than 100 seats" where $T_n = 12 + 2(n - 1)$, solve $12 + 2(n - 1) > 100$: $n > 45$, so row 46. For a sum, solve the quadratic $S_n = $ target, then take the next whole number. Check the answer and the one before it.

**Splitting a sum.** $(2 + 1) + (4 + 1) + (8 + 1) + \dots$ for 10 terms is a geometric series $2 + 4 + 8 + \dots$ plus ten 1s: $\dfrac{2(2^{10} - 1)}{2 - 1} + 10 = 2046 + 10 = 2056$.

## Quick summary

- An arithmetic sequence adds the same amount $d$ each step.
- $T_n = a + (n - 1)d$, where $a$ is the first term.
- The graph of $T_n$ against $n$ is a straight line with gradient $d$.
- $S_n = \dfrac{n}{2}(a + l) = \dfrac{n}{2}[2a + (n - 1)d]$.
- $\sum$ means add up the terms.

## Quiz

Try these questions from the chapter quiz.

1. What is the next term of 7, 11, 15, 19, ...?
   - A. 22
   - B. 23
   - C. 24
   - D. 21
   - Answer: B. 23. Add 4 each time.

2. What is the common difference of 20, 17, 14, 11, ...?
   - A. 3
   - B. $-3$
   - C. 17
   - D. $-17$
   - Answer: B. $-3$. Each term is 3 less.

3. A pattern has rule $T_n = 5n - 2$. What is $T_6$?
   - A. 30
   - B. 28
   - C. 32
   - D. 23
   - Answer: B. 28. $5(6) - 2 = 28$.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/mathematics-advanced/arithmetic-sequences/#quiz