Every equation you've solved so far described one moment in time. But many real patterns keep going — Copper's weight each week, Mia's savings each month, the number of jumps in each training set. When a list of numbers grows (or shrinks) by the same amount each time, it's called an arithmetic sequence. The fixed amount you add each time is the common difference.
Vocabulary
Term ($a_n$): one number in the sequence. The 1st term is $a_1$, the 5th is $a_5$. Common difference ($d$): the fixed amount added each step. Can be positive (growing) or negative (shrinking). Position ($n$): which term it is (1st, 2nd, 3rd …).
Formula for the $n$th term:
$$a_n = a_1 + (n - 1) \cdot d$$
Start at $a_1$, then add $d$ exactly $(n-1)$ more times.
Worked Examples
Worked Example 1 — identify and extend
Copper's training schedule adds 3 extra minutes of trot each week. Week 1: 12 min. List the first 5 weeks and find the 10th term.
A show-jumping course has a sequence of obstacle heights (in cm): 65, __, 83, 92, …
The difference between $a_3 = 83$ and $a_4 = 92$ gives $d = 9$. So $a_2 = 83 - 9 = $ 74 cm. Check: $65 + 9 = 74$ ✓.
Worked Example 4 — write the expression
A sequence starts at 7 and has common difference $-5$. Write a formula for the $n$th term and find the first negative term.
$$a_n = 7 + (n-1)(-5) = 7 - 5n + 5 = 12 - 5n$$
When does $a_n < 0$? $12 - 5n < 0 \;\Rightarrow\; n > 2.4$, so the 3rd term ($n=3$): $12 - 15 = -3$. ✓
Practice Problems
Problems 1–3: identify the common difference and fill in the missing term(s).
Feeding routine. Mia measures how many kg of hay Copper eats per day during different seasons. The amounts follow a sequence: 7, 9, 11, __, 15, __
$d = $
Missing terms:
Stable rent. The monthly stable rent increases each year: $\$520,\ \$547,\ \__,\ \$601,\ \$628$
$d = $
Missing term:
Training load. Copper starts with 18 minutes of canter per session. Each week the trainer adds 2.5 minutes.
(a) Write the formula for $a_n$.
(b) How long will the canter session be in week 11?
Prize money. A local show offers prize money in arithmetic sequence by placing: 1st place 120 €, 2nd place 103 €, 3rd place 86 €, …
(a) What is $d$?
(b) What does 8th place receive?
(c) Which placing first receives $0$ € or less?
Savings plan. Mia has $\$35$ saved. She adds $\$22$ each week.
(a) Write the formula for $a_n$ (week $n$, counting week 1 as when she already has $\$35$).
(b) After how many weeks does she first have at least $\$200$?
From here — write the sequence and expression yourself.
Bale countdown. Mia starts the month with 53 kg of hay. Copper eats 7 kg per day.
Write out the first 5 terms of the sequence (day 1 = start of day, so $a_1 = 53$).
Write the formula for $a_n$:
On which day does she fall below 10 kg and need to reorder?
Jump height progression. At a show, the 1st fence is 55 cm. Each fence is 8 cm higher than the previous one.
Write the expression $a_n = $ ____
The maximum safe height for Copper is 135 cm. What is the last fence number he can safely attempt?
Vet bills. Routine vet visits cost 45 €, 62 €, 79 €, 96 € over 4 visits.
Is this arithmetic? Show how you know.
Predict the 9th visit cost.
Challenge. The 3rd term of an arithmetic sequence is 29, and the 7th term is 53. Find $d$ and $a_1$, then write the $n$th term formula. (Hint: how many steps separate the 3rd and 7th terms?)
What's next → Arithmetic sequences add a fixed amount each step. But what if you multiply by a fixed amount instead? Doubling every day, halving every week — these are geometric sequences. Lesson 09 covers how to spot them, find the ratio, and write the formula.