What is a geometric sequence?

A list of numbers where you multiply by the same amount, the common ratio r, to get from each term to the next. 3, 6, 12, 24 is geometric with r = 2.

What is the formula for the nth term?

tₙ = arⁿ⁻¹, where a is the first term. Some textbooks write aₙ instead of tₙ.

How do I find which term a number is?

Divide it by a, then find the power of r that gives the result. In 5, 10, 20, …, 640 ÷ 5 = 128 = 2⁷, so n − 1 = 7 and 640 is t₈.

What happens when r is negative or a fraction?

A negative r makes the signs alternate: 2, −6, 18, −54. A ratio between 0 and 1 makes the terms shrink: 486, 162, 54, 18.

Geometric Sequences · Grades 11–11

Geometric Sequences Worksheets

Grade 11 geometric sequences in Ontario notation, one kind only: find the common ratio, write the general term and recursive formula, read terms from a table or graph, and handle negative and fraction ratios. Every answer key shows the working.

Generate geometric sequences worksheets →

Free plan · 2 credits every month · answer keys included · no card required.

What it is

A geometric sequence multiplies by the same number, the common ratio r, to get from each term to the next. a is the first term and $t_{n}$ is the term in position n. Plotted against n, the terms curve, so a geometric sequence is an exponential function of n.

Why it matters

A geometric sequence is an exponential function whose inputs are 1, 2, 3, and so on. Compound interest, population growth and half-life all follow the same pattern, so this is the start of the exponential unit.

How to do it

Find r: divide any term by the one before it. Check that every ratio is the same.
General term: $t_{n} = ar^{n-1}$ .
To find a term, substitute its position for n. Work out the power first, then multiply by a.
To find which term a value is, divide it by a, then find the power of r that gives the result.
Recursive formula: $t_{1} = a$ , $t_{n} = rt_{n-1}$ .

Examples by level

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

Beginner $t_{1} = 7$ , $t_{n} = 4t_{n-1}$ . What are the first four terms?Answer7, 28, 112, 448
IntermediateWhat is a recursive formula for 5, 15, 45, 135, …?Answer $t_{1} = 5$ , $t_{n} = 3t_{n-1}$
AdvancedHow many terms are in the sequence 3, 9, 27, …, 243?Answer5 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 as the exponent in $t_{n} = ar^{n-1}$ .
Multiplying a by r before taking the power: $3(2)^{7}$ is 3 × 128, not $6^{7}$ .
Dropping the brackets on a negative ratio: $(-2)^{4} = 16$ , but $-2^{4} = -16$ .
Adding the common ratio instead of multiplying by it.

Tips

The exponent is n − 1, not n: the first term has not been multiplied yet.
Work out the power before multiplying by a: $3(2)^{7}$ is 3 × 128, not $6^{7}$ .
Keep brackets on a negative r: $(-2)^{4} = 16$ , but $-2^{4} = -16$ .
A negative r makes the signs alternate. An r between 0 and 1 makes the terms shrink.

For parents

Ask: what does each term get multiplied by? That number is r. If it is negative, the signs flip back and forth. If it is a fraction, the terms shrink.

For teachers

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

Key vocabulary

common ratio: r, the number each term is multiplied by
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

Frequently asked questions

What is a geometric sequence?: A list of numbers where you multiply by the same amount, the common ratio r, to get from each term to the next. 3, 6, 12, 24 is geometric with r = 2.
What is the formula for the nth term?: $t_{n} = ar^{n-1}$ , where a is the first term. Some textbooks write $a_{n}$ instead of $t_{n}$ .
How do I find which term a number is?: Divide it by a, then find the power of r that gives the result. In 5, 10, 20, …, 640 ÷ 5 = 128 = $2^{7}$ , so n − 1 = 7 and 640 is $t_{8}$ .
What happens when r is negative or a fraction?: A negative r makes the signs alternate: 2, −6, 18, −54. A ratio between 0 and 1 makes the terms shrink: 486, 162, 54, 18.

Ready to practice?

Generate a print-ready geometric sequences pack with answer keys: pick a grade and difficulty, and it's ready in seconds.

Create free worksheets

Free to start · answer keys included · free accounts get 14 days of Standard · no card required.