C. Discrete Functions · Sequences and Series

Arithmetic, Geometric or Neither?

C2.1 C1.4 C1.1About 65 minutes

Learning goals

I can classify a sequence as arithmetic, geometric or neither from its terms.
I can classify a sequence from its explicit formula or its recursion formula.
I can prove that a sequence is arithmetic or geometric using tₙ − tₙ₋₁ or tₙ ÷ tₙ₋₁.
I can connect each type to the shape of its graph.

Words to know

classify
Sort into a category, here arithmetic, geometric, both or neither, with a reason.

constant sequence
A sequence whose terms are all equal, such as 7, 7, 7, …. If the terms are not 0, it is both arithmetic (d = 0) and geometric (r = 1).

harmonic sequence
The sequence 1, 1/2, 1/3, 1/4, …. It is neither arithmetic nor geometric.

Sorting the sequences

You now know two families of sequences: arithmetic (add the same number each time) and geometric (multiply by the same number each time). Plenty of sequences belong to neither family: the triangular numbers, the square numbers and the Fibonacci numbers, for example. Before you can use a formula, you must know which family a sequence belongs to. This lesson is about classifying sequences quickly and correctly, whether they are given as a list of terms, an explicit formula, a recursion formula or a graph.

A decision chart for classifying a sequence from its terms. Test the differences first, then the ratios.

Worked example

Example 1: Classifying from terms

Classify each sequence as arithmetic, geometric or neither.

(a) 4, 7, 10, 13, … (b) 4, 8, 16, 32, … (c) 4, 7, 11, 16, … (d) 1, 1/2, 1/3, 1/4, … (e) 6, 6, 6, 6, …

(a) Differences 3, 3, 3. Arithmetic, d = 3.
(b) Differences 4, 8, 16 (not constant). Ratios 2, 2, 2. Geometric, r = 2.
(c) Differences 3, 4, 5 (not constant). Ratios 7/4, 11/7, … (not constant). Neither. (The second differences are constant, so it is quadratic.)
(d) Differences −1/2, −1/6, −1/12 and ratios 1/2, 2/3, 3/4: neither is constant. Neither.
(e) Differences 0, 0, 0 and ratios 1, 1, 1. Both arithmetic (d = 0) and geometric (r = 1).

(a) arithmetic (b) geometric (c) neither (d) neither (e) both

Classifying from an explicit formula

The formula often tells you the type at a glance:

Linear in n (like tₙ = 5 − 2n or tₙ = 3n + 8): arithmetic, with d equal to the coefficient of n.
A constant times a power with n in the exponent (like tₙ = 3(2)ⁿ or tₙ = 4/3ⁿ): geometric, with r equal to the base.
Anything else (n², 1/n, n(n + 1)/2, …): usually neither. Check a few terms to be sure.

To prove the type, simplify tₙ − tₙ₋₁ or tₙ ÷ tₙ₋₁. If the result does not depend on n, it is a constant d or r.

Worked example

Example 2: Proving the type

Prove that tₙ = 5 − 2n is arithmetic and that tₙ = 3(2)ⁿ is geometric. Show that tₙ = n² + 1 is neither.

tₙ = 5 − 2n: tₙ − tₙ₋₁ = (5 − 2n) − (5 − 2(n − 1)) = 5 − 2n − 5 + 2n − 2 = −2, a constant. Arithmetic with d = −2.
tₙ = 3(2)ⁿ: tₙ ÷ tₙ₋₁ = 3(2)ⁿ ÷ 3(2)ⁿ⁻¹ = 2¹ = 2, a constant. Geometric with r = 2.
tₙ = n² + 1: terms 2, 5, 10, 17. Differences 3, 5, 7 and ratios 2.5, 2, 1.7. Neither is constant, so it is neither. (Algebraically, tₙ − tₙ₋₁ = 2n − 1 depends on n.)

arithmetic (d = −2); geometric (r = 2); neither

Worked example

Example 3: Classifying from a recursion formula

Classify: (a) t₁ = 9, tₙ = tₙ₋₁ − 7 (b) t₁ = 40, tₙ = −0.5tₙ₋₁ (c) t₁ = 1, tₙ = 2tₙ₋₁ + 1 (d) t₁ = 1, tₙ = tₙ₋₁ + n

(a) "Previous term plus a constant": arithmetic, d = −7.
(b) "Previous term times a constant": geometric, r = −0.5.
(c) Multiply and add: terms 1, 3, 7, 15, 31. Differences 2, 4, 8, … and ratios 3, 2.33, … are not constant: neither.
(d) The amount added changes with n: terms 1, 3, 6, 10. Neither (these are the triangular numbers).

(a) arithmetic (b) geometric (c) neither (d) neither

Arithmetic and geometric sequences overlap only in the constant sequences (with non-zero terms). Many important sequences sit outside both families.

Worked example

Example 4: Algebraic terms

Classify (a) 2x, 6x², 18x³, … (x ≠ 0) (b) a, a + b, a + 2b, … (c) x, 2x, 4x + 1, …

(a) Ratios: 6x² ÷ 2x = 3x and 18x³ ÷ 6x² = 3x. Geometric, r = 3x.
(b) Differences: b and b. Arithmetic, d = b.
(c) Differences: x and 2x + 1, equal only if x = −1. Ratios: 2 and (4x + 1)/(2x), equal only if 4x + 1 = 4x, which never happens. In general neither (it is arithmetic only in the special case x = −1: −1, −2, −3).

(a) geometric (b) arithmetic (c) neither, except arithmetic when x = −1

Type	Test	Explicit formula	Recursion	Graph
Arithmetic	constant difference d	tₙ = a + (n − 1)d (linear in n)	tₙ = tₙ₋₁ + d	points on a line
Geometric	constant ratio r	tₙ = arⁿ⁻¹ (exponential in n)	tₙ = r·tₙ₋₁	points on an exponential curve (alternating if r < 0)
Neither	neither is constant	anything else	anything else	other patterns

Summary of the three cases.

Remember from earlier

This is the same thinking you used in the Exponential Functions unit to tell linear, quadratic and exponential functions apart with finite differences and ratios, and in Grade 10 when constant second differences told you a relation was quadratic. A sequence with constant second differences is quadratic, which means it is neither arithmetic nor geometric.

Just for fun

Why is the sequence 5, 5, 5, 5, … so popular?

It gets invited to both the arithmetic party and the geometric party, and it behaves the same at each one.

Hands-on activity

Sequence sorting cards

You need: 12 small pieces of paper, a pencil and a calculator.

Write one sequence on each card:

1, 3, 5, 7 | 2, 6, 18, 54 | 1, 4, 9, 16 | 100, 90, 80, 70 | 1, 1, 2, 3, 5 | 81, 27, 9, 3 | 3, 3, 3, 3 | 1, 2, 4, 7 | −2, 4, −8, 16 | 0.5, 1, 1.5, 2 | 1, 1/2, 1/4, 1/8 | 2, 3, 5, 8

Sort them into four piles: arithmetic, geometric, both, neither. For every arithmetic or geometric card, write the general term on the back. Ask someone at home to shuffle and time you on a second try.

Answer key: Arithmetic: 1, 3, 5, 7 (tₙ = 2n − 1); 100, 90, 80, 70 (tₙ = 110 − 10n); 0.5, 1, 1.5, 2 (tₙ = 0.5n). Geometric: 2, 6, 18, 54 (tₙ = 2(3)ⁿ⁻¹); 81, 27, 9, 3 (tₙ = 81(1/3)ⁿ⁻¹); −2, 4, −8, 16 (tₙ = −2(−2)ⁿ⁻¹ = (−2)ⁿ); 1, 1/2, 1/4, 1/8 (tₙ = (1/2)ⁿ⁻¹). Both: 3, 3, 3, 3. Neither: 1, 4, 9, 16; 1, 1, 2, 3, 5; 1, 2, 4, 7; 2, 3, 5, 8.

Your turn

Classify 12, 9, 6, 3, ….

Classify 2, −10, 50, −250, ….

Classify 1, 8, 27, 64, ….

Classify tₙ = 7n − 4 and tₙ = 7(4)ⁿ.

Classify t₁ = 3, tₙ = 4tₙ₋₁ − 3.

Prove that tₙ = 10(0.6)ⁿ is geometric.

More practice: warm-up

Classify 5, 10, 15, 20, ….

Classify 5, 10, 20, 40, ….

Classify 5, 10, 16, 23, ….

Classify t₁ = 2, tₙ = tₙ₋₁ + 0.5.

Classify t₁ = 2, tₙ = 0.5tₙ₋₁.

More practice: on your own

Prove that tₙ = 4 + 3n is arithmetic.

Classify tₙ = 2ⁿ + 1. Explain.

Classify √2, 2, 2√2, 4, ….

Classify 0.1, 0.01, 0.001, ….

Classify x + 1, x + 4, x + 7, … and find its general term.

Challenge

Can a sequence with a term equal to 0 be geometric? Explain.

Find all values of x for which 4, x, 9 is (a) arithmetic, (b) geometric.

A sequence is both arithmetic and geometric. Prove that all its terms are equal.

Check your understanding

Quiz

The sequence 3, 12, 48, 192, … is:

The sequence 2, 5, 9, 14, … is:

Which formula gives an arithmetic sequence?

Which recursion formula gives a geometric sequence?

Which sequence is both arithmetic and geometric?

For tₙ = 6(0.5)ⁿ, tₙ ÷ tₙ₋₁ equals:

For parents and teachers

Big idea: Expectation C2.1 asks students to identify sequences as arithmetic, geometric or neither, given a numeric or algebraic representation. This lesson practises all of them: lists of terms, explicit formulas, recursion formulas and algebraic terms.

How to help: Write a few sequences and ask "add or multiply?" Then ask for proof: "Show me the differences" or "Show me the ratios". Encourage checking at least three pairs of terms.

Common difficulties:

Deciding from only two terms. Any two terms have a difference and a ratio.
Thinking "increasing faster and faster" means geometric. Quadratic sequences also speed up; check the ratios.
Forgetting the "both" case (constant sequences).
Recursion formulas like tₙ = 2tₙ₋₁ + 1 that multiply and add are neither.