Arithmetic & Geometric Sequences

You’ll be able to

Write explicit and recursive formulas for arithmetic and geometric sequences
Identify which sequence type a table of values represents
Connect arithmetic sequences to linear functions and geometric sequences to exponential functions

Adding versus multiplying

An arithmetic sequence adds a fixed common difference d each step. A geometric sequence multiplies by a fixed common ratio r each step. That is the entire distinction, and it is the same distinction as linear versus exponential — sequences are simply those functions restricted to whole-number inputs. Recognizing which operation is repeating is the first move in every sequence problem.

Explicit formulas

arithmetic: aₙ = a₁ + (n − 1)d · geometric: gₙ = g₁ · r^(n−1)

The exponent and the multiplier are (n − 1), not n, when indexing from a₁. Off-by-one here is the most common error in the topic.

Recursive form says the same thing differently

A recursive definition gives the first term and a rule for getting the next one from the current one: aₙ = aₙ₋₁ + d, or gₙ = r·gₙ₋₁. Recursion describes the process and is often how a situation is naturally stated ("each year the deer population grows 8%"). The explicit formula describes the result and is what you need to reach the 40th term without computing the 39 before it. Fluency means translating between them.

Worked example

A sequence begins 7, 11, 15, 19, … Find explicit and recursive formulas, then the 30th term.

1.Differences: 11 − 7 = 4, 15 − 11 = 4, 19 − 15 = 4. Constant, so arithmetic with d = 4.
2.First term a₁ = 7.
3.Explicit: aₙ = 7 + (n − 1)(4) = 4n + 3.
4.Recursive: a₁ = 7 and aₙ = aₙ₋₁ + 4 for n ≥ 2.
5.a₃₀ = 4(30) + 3 = 123. Or via the unsimplified form: 7 + 29(4) = 7 + 116 = 123. ✓

Answer: aₙ = 4n + 3 explicitly, or a₁ = 7 with aₙ = aₙ₋₁ + 4 recursively. The 30th term is 123. Note that simplifying to 4n + 3 makes the connection to the line y = 4x + 3 immediate.

Checkpoint

A geometric sequence has g₁ = 5 and g₄ = 40. What is the common ratio?

Watch out

Check both differences and ratios before deciding. The sequence 2, 4, 6 has constant difference 2 and non-constant ratios, so it is arithmetic; 2, 4, 8 has constant ratio 2 and non-constant differences, so it is geometric. Only one test passing is what identifies the type.

Sequences are functions with a restricted domain

Plot aₙ = 4n + 3 for n = 1, 2, 3, … and you get points on the line y = 4x + 3 — the same rule, sampled at whole numbers. Likewise gₙ = 5·2^(n−1) samples the exponential y = 2.5·2ˣ. This is why the differences-versus-ratios test from function modeling works identically on sequences: linear growth adds a constant amount, exponential growth multiplies by a constant factor, and a sequence is just that idea one step at a time.

Worked example

A car worth $28,000 loses 15% of its value each year. Write a sequence for its value at the start of year n and find the value at the start of year 6.

1.Losing 15% leaves 85%, so each year the value is multiplied by 0.85. Geometric with r = 0.85.
2.Start of year 1 is the purchase moment: g₁ = 28000.
3.Explicit: gₙ = 28000·(0.85)^(n−1).
4.Start of year 6 means n = 6, so the exponent is 5: g₆ = 28000·(0.85)⁵.
5.(0.85)⁵ ≈ 0.4437, so g₆ ≈ 28000(0.4437) ≈ 12,424.

Answer: gₙ = 28000·(0.85)^(n−1), and the value at the start of year 6 is about $12,424. Depreciation of a fixed percentage is geometric; depreciation of a fixed dollar amount would be arithmetic, and the two diverge sharply over time.

Checkpoint

Which sequence is neither arithmetic nor geometric?

Answer the 2 checkpoints as you read.