What is the difference between arithmetic and geometric sequences?

Arithmetic sequences add a constant, the common difference d, each term. Geometric sequences multiply by a constant, the common ratio r.

What is the formula for the nth term?

Arithmetic: tₙ = a + (n − 1)d. Geometric: tₙ = arⁿ⁻¹. Here a is the first term. Some textbooks write aₙ instead of tₙ.

How do I find which term a number is?

Write the general term, set it equal to the number, and solve for n. In 5, 9, 13, …, tₙ = 4n + 1, so 81 is t₂₀.

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.

What is a recursive formula?

It gives the first term and the rule from one term to the next, like t₁ = 5, tₙ = tₙ₋₁ + 4. An explicit formula like tₙ = 4n + 1 jumps straight to any term.

Sequences · Grades 11–11

Sequences Worksheets

Grade 11 mixed review of arithmetic and geometric sequences in Ontario notation: classify a sequence, write its general term and recursive formula, read terms from a table or graph, find any term, and solve backwards for which term or for a and d. Every answer key shows the working. For one kind at a time, use the Arithmetic Sequences or Geometric Sequences worksheets.

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What it is

A sequence is an ordered list of terms. $t_{n}$ is the term in position n, and a is the first term, $t_{1}$ . An arithmetic sequence adds the same common difference d each time, so it grows like a linear function. A geometric sequence multiplies by the same common ratio r each time, so it behaves like an exponential function.

Why it matters

Arithmetic sequences are linear functions sampled at whole numbers, and geometric sequences are exponential ones. Seeing that link here makes compound interest, annuities and exponential growth feel familiar later.

How to do it

Decide the type. Subtract neighbouring terms: all the same means arithmetic. If not, divide them: all the same means geometric. Otherwise it is neither.
Arithmetic general term: $t_{n} = a + (n - 1)d$ , then simplify.
Geometric general term: $t_{n} = ar^{n-1}$ .
To find a term, substitute its position for n. To find which term a value is, set the formula equal to the value and solve for n.
A recursive formula gives the first term and the rule from one term to the next, like $t_{1} = 5$ , $t_{n} = t_{n-1} + 4$ .

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 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.
Dropping the brackets on a negative ratio: $(-2)^{4} = 16$ , but $-2^{4} = -16$ .
Calling 1, 4, 9, 16 arithmetic. Its differences (3, 5, 7) change, so it is neither.
Adding the common ratio instead of multiplying by it.

Tips

Use n − 1, not n: the first term has had no differences added yet.
Keep brackets on a negative d or 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.
1, 4, 9, 16 is neither. Its second differences (2, 2) are constant: a quadratic.

For parents

Ask: what do you do to get from one term to the next? Adding the same amount is arithmetic; multiplying by the same amount is geometric.

For teachers

Beginner sheets classify, generate terms and use f(n) notation. Intermediate sheets write and use formulas, including from a table or a graph. Advanced sheets solve backwards, count terms, use second differences and include short word problems. Each sheet mixes question types, and the instruction line says so.

Key vocabulary

general term: the formula for $t_{n}$ ; some textbooks write $a_{n}$
common difference: d, the amount added each time in an arithmetic sequence
common ratio: r, the number each term is multiplied by in a geometric sequence
recursive formula: the first term plus a rule that builds each term from the one before
explicit formula: a formula that gives any term directly from its position n

Frequently asked questions

What is the difference between arithmetic and geometric sequences?: Arithmetic sequences add a constant, the common difference d, each term. Geometric sequences multiply by a constant, the common ratio r.
What is the formula for the nth term?: Arithmetic: $t_{n} = a + (n - 1)d$ . Geometric: $t_{n} = ar^{n-1}$ . Here a is the first term. Some textbooks write $a_{n}$ instead of $t_{n}$ .
How do I find which term a number is?: Write the general term, set it equal to the number, and solve for n. In 5, 9, 13, …, $t_{n} = 4n + 1$ , so 81 is $t_{20}$ .
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.
What is a recursive formula?: It gives the first term and the rule from one term to the next, like $t_{1} = 5$ , $t_{n} = t_{n-1} + 4$ . An explicit formula like $t_{n} = 4n + 1$ jumps straight to any term.

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