What is an arithmetic sequence?

A list of numbers where you add the same amount, the common difference d, to get from each term to the next. 5, 9, 13, 17 is arithmetic with d = 4.

What is the formula for the nth term?

tₙ = a + (n − 1)d, where a is the first term. Some textbooks write aₙ instead of tₙ.

How do I find how many terms a sequence has?

Set the general term equal to the last term and solve for n. In 6, 11, 16, …, 201, 201 = 6 + (n − 1)(5) gives n = 40.

How do I find a and d from two terms?

Subtract the two terms and divide by the number of steps between them to get d, then work back to the first term. If t₅ = 17 and t₁₂ = 45, then 7d = 28, so d = 4 and a = 1.

Arithmetic Sequences · Grades 11–11

Arithmetic Sequences Worksheets

Grade 11 arithmetic sequences in Ontario notation, one kind only: find the common difference, write the general term and recursive formula, read terms from a table or graph, and solve backwards for which term, how many terms, or a and d. Every answer key shows the working.

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What it is

An arithmetic sequence adds the same number, the common difference d, to get from each term to the next. $t_{n}$ is the term in position n, and a is the first term. Plotted against n, the terms lie on a straight line, so an arithmetic sequence is a linear function of n.

Why it matters

An arithmetic sequence is a linear function whose inputs are 1, 2, 3, and so on: d is the slope. Seeing that link makes simple interest, linear growth and the arithmetic series that follow feel familiar.

How to do it

Find d: subtract any term from the one after it. Check that every gap is the same.
General term: $t_{n} = a + (n - 1)d$ , then simplify.
To find a term, substitute its position for n.
To find which term a value is, or how many terms a list has, set the general term equal to the value and solve for n.
Recursive formula: $t_{1} = a$ , $t_{n} = t_{n-1} + d$ .

Examples by level

Read each question carefully. They're not all the same kind of question.

Beginner $t_{1} = 7$ , $t_{n} = t_{n-1} - 5$ . What are the first four terms?Answer7, 2, −3, −8
IntermediateWhat is a recursive formula for 0, −6, −12, −18, …?Answer $t_{1} = 0$ , $t_{n} = t_{n-1} - 6$
AdvancedHow many terms are in the sequence 5, 2, −1, …, −127?Answer45 terms

Examples are generated by the same engine as the worksheets, so they're always mathematically correct. Built to avoid repeats, not reshuffle the same handful of questions.

Common mistakes

Using n instead of n − 1 in $t_{n} = a + (n - 1)d$ : the first term has had no differences added yet.
Subtracting in the wrong order, which flips the sign of d. Always take a term minus the one before it.
Losing the brackets on a negative d: 7 + (n − 1)(−4), not 7 + n − 1 − 4.
Counting the gaps instead of the terms in "how many terms" questions: solve for n, not n − 1.

Tips

Use n − 1, not n: the first term has had no differences added yet.
A negative d means the terms go down. Keep its brackets: 7 + (n − 1)(−4).
Check a formula by substituting n = 1. You should get the first term.
On a graph the points lie on a line, and d is its slope.

For parents

Ask: what gets added each time? That number is d. Then ask how many times it has been added by the 10th term. The answer is 9, which is where n − 1 comes from.

For teachers

Every question on these sheets is arithmetic, for the day the class meets them. Beginner sheets find d and generate terms. Intermediate sheets write and use formulas, including from a table or a graph. Advanced sheets solve backwards and include short word problems. For a mixed review that also asks students to classify, use the Sequences worksheets.

Key vocabulary

common difference: d, the amount added each time
general term: the formula for $t_{n}$ ; some textbooks write $a_{n}$
recursive formula: the first term plus a rule that builds each term from the one before
function notation: f(n), another way to write $t_{n}$

Frequently asked questions

What is an arithmetic sequence?: A list of numbers where you add the same amount, the common difference d, to get from each term to the next. 5, 9, 13, 17 is arithmetic with d = 4.
What is the formula for the nth term?: $t_{n} = a + (n - 1)d$ , where a is the first term. Some textbooks write $a_{n}$ instead of $t_{n}$ .
How do I find how many terms a sequence has?: Set the general term equal to the last term and solve for n. In 6, 11, 16, …, 201, 201 = 6 + (n − 1)(5) gives n = 40.
How do I find a and d from two terms?: Subtract the two terms and divide by the number of steps between them to get d, then work back to the first term. If $t_{5} = 17$ and $t_{12} = 45$ , then 7d = 28, so d = 4 and a = 1.

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