Feed a function only the whole numbers 1, 2, 3, … and its outputs line up into a sequence.
Count out loud: 1, 3, 5, 7, 9, … — the odd numbers, in their natural order. You have just written a sequence. Behind it sits a single rule, f(n) = 2n − 1; feed that rule the position numbers 1, 2, 3, 4, … and out come the terms one after another. A sequence, then, is nothing exotic: it is an ordinary function whose inputs are restricted to the whole numbers, so its graph is not a smooth curve but a line of dots, one per position. In this lesson we meet the vocabulary — terms, the index n, the general term aₙ — and the two ways to pin a sequence down: name each term outright with a formula, or tell each term how to grow from the one before.
A sequence is a list of numbers written in a fixed order. We write it inside braces with a general name: {aₙ}. The numbers themselves are the terms, and a sequence is exactly its terms in their order — the same numbers shuffled into a different order are a different sequence.
That last point is the whole personality of a sequence. The list 2, 4, 6 and the list 4, 2, 6 contain the same three numbers, yet they are not equal as sequences: position matters. A set {2, 4, 6} forgets order; a sequence never does. (This is why we say a sequence carries order as well as size — a hint we will cash in at the very end of Stage 26 when a number is about to gain a direction and become a vector.)
A sequence can stop or run forever. A list with a last term — say the five test scores 72, 85, 85, 90, 96 — is a finite sequence. A list that never ends — the odd numbers 1, 3, 5, 7, …, where the "…" means keep going — is an infinite sequence. Most of the interesting questions in this stage are about infinite sequences: what they head toward, and how to add up the first n of them.
A sequence is an ordered list of numbers, written {aₙ}. Order is part of its identity: 2, 4, 6 ≠ 4, 2, 6. It is finite if it has a last term, infinite if it goes on forever.
Each term sits in a numbered slot. The first term is a₁, the second is a₂, and the term in slot number n — the general term — is aₙ. The little number written below is the index (or subscript): it is the position, not the value. So in 1, 3, 5, 7, … we have a₁ = 1, a₂ = 3, a₃ = 5, a₄ = 7.
Keep the two roles straight. The index n answers "which slot?"; the term aₙ answers "what number is in it?". They almost never agree: in the odd-number sequence, slot n = 4 holds the value a₄ = 7, not 4. The index is the input; the term is the output.
For the sequence 3, 6, 9, 12, 15, … of multiples of 3: a₁ = 3, a₃ = 9, and the term in slot 10 is a₁₀ = 30. The index 10 and the term 30 are different numbers — one names the place, the other names the value.
Now the unifying idea. A function takes an input and returns one output. A sequence does exactly that — its input is the position n and its output is the term aₙ — with one restriction: the inputs are only the positive integers, ℕ⁺ = {1, 2, 3, …}. So we may honestly write
aₙ = f(n), with domain n ∈ ℕ⁺.
This single restriction changes everything about the picture. An ordinary function like f(x) = 2x − 1 accepts every real x — 1.5, π, −3 — and its graph is an unbroken line. But a sequence accepts only the whole numbers, so we plot a dot at n = 1, a dot at n = 2, a dot at n = 3 — and nothing in between. There is no such thing as the "1.5-th term." The graph of a sequence is therefore a row of isolated dots, never a continuous curve.
A sequence is not a continuous curve. The dots may land on a line or curve — that underlying rule is drawn faint blue — but the sequence itself is the isolated dots only. Never connect the dots. There is no value between two terms, because there is no whole number between n = 3 and n = 4.
The first widget lets you choose a rule aₙ = f(n) and watch the function turn into a row of dots. Notice the readout: it lists a₁ through a₈ and reports whether the dots are climbing, sinking, or bouncing.
Pick a formula for aₙ. We plug in n = 1, 2, …, 8 and plot the terms as dots — never a line.
The most direct way to define a sequence is to name every term at once with a formula for aₙ in terms of the index n. Then you read off any term you like by substituting: want the 100th term? Plug in n = 100. This expression is the general-term formula (or explicit formula). Three workhorses:
| General term | n = 1 | n = 2 | n = 3 | n = 4 | The sequence |
|---|---|---|---|---|---|
| aₙ = 2n − 1 | 1 | 3 | 5 | 7 | the odd numbers |
| aₙ = n2 | 1 | 4 | 9 | 16 | the perfect squares |
| aₙ = (−1)n | −1 | 1 | −1 | 1 | alternating ±1 |
Each one is a tiny machine. For aₙ = n2 the work is just arithmetic: a₁ = 12 = 1, a₂ = 22 = 4, a₃ = 32 = 9, a₄ = 42 = 16. For aₙ = (−1)n the power of −1 flips sign with each step — even n gives +1, odd n gives −1 — so the dots oscillate forever between two heights. The factor (−1)n is the standard switch for "alternate the sign."
Find the 10th term of aₙ = n2 − n. Substitute n = 10: a₁₀ = 102 − 10 = 100 − 10 = 90. No need to list the first nine — the formula gives any term in one step. That is the power of an explicit rule.
The first few terms do not determine the rule. The start 2, 4, 8, … looks like doubling, aₙ = 2n−1 — but it could just as well continue 2, 4, 8, 14, 22, … (adding 2, 4, 6, 8, …), which is aₙ = n2 − n + 2. Reading a rule off a handful of terms is a guess, not a proof. A sequence is fixed only once its general term or recurrence is given.
There is a second way to pin a sequence down. Instead of naming the value in slot n outright, you say how to get each term from the one before it — and where to start. That pair of instructions is a recurrence relation. A classic:
a₁ = 2, aₙ₊₁ = aₙ + 3.
Read it as a recipe. Start at 2. To get the next term, add 3: 2 → 5 → 8 → 11 → 14 → …. The first line is the base (without a starting value the recipe has nothing to chew on); the second line is the step. Together they grow the whole sequence, one term at a time.
An explicit formula and a recurrence are two descriptions of the same object, with a real trade-off:
| What it tells you | To get a₅₀ | |
|---|---|---|
| Explicit aₙ = f(n) | the value in any slot, directly | one substitution: a₅₀ = f(50) |
| Recurrence aₙ₊₁ = g(aₙ) | how each term grows from the last | grind out all 49 steps first |
The recurrence is often how a process naturally arises — a bank balance is born from last month's balance, a population from last year's — while the explicit formula is what you want for a distant term. Much of Stage 26 is the art of turning one into the other.
The second widget is a recurrence machine: set the start a₁ and the step d, then build the terms one at a time and watch the dots appear.
Set the start a₁ and the step d in the rule aₙ₊₁ = aₙ + d. We grow the terms with recurVals and plot them as dots.
Once the terms are dots, their shape tells a story at a glance. Compare each term with the next:
You read this straight off the picture. The squares 1, 4, 9, 16, … climb, and climb faster and faster (they sit on a parabola). The halvings 8, 4, 2, 1, … sink toward 0. The alternator −1, 1, −1, 1, … never settles — it oscillates. The general-term explorer above already announces which of the three is happening; try each formula and confirm it by eye.
A sequence is a function on ℕ⁺, so it lives as a row of dots. Compare aₙ with aₙ₊₁ to call it increasing, decreasing, or oscillating — no calculation of a slope needed, just neighbour-to-neighbour comparison.
A sequence is a function fed only the whole numbers, so it shows up as a line of dots — and you can describe it in two complementary ways.
| Idea | Symbol / form | What it means |
|---|---|---|
| Sequence | {aₙ} | an ordered list; 2,4,6 ≠ 4,2,6 |
| Term & index | aₙ at position n | n = the slot (input), aₙ = the value (output) |
| A special function | aₙ = f(n), n ∈ ℕ⁺ | domain = positive integers → graph is dots, not a curve |
| General term | aₙ = expression in n | any term in one substitution: aₙ = n² → 1,4,9,16 |
| Recurrence | a₁ + aₙ₊₁ = g(aₙ) | a base term + a step; a₁=2, aₙ₊₁=aₙ+3 → 2,5,8,11 |
| Behaviour | compare aₙ, aₙ₊₁ | climbing / sinking / oscillating — read off the dots |
Two traps to remember: a sequence is never a connected curve, and the first few terms do not determine the rule. Next, in 26.2 Arithmetic sequences, we add the same step d every time — and the dots fall in a perfectly straight line.
For the sequence aₙ = 3n + 1, write out a₁, a₂, a₃, a₄, and find a₂₀.
Substitute each index. a₁ = 3·1 + 1 = 4, a₂ = 7, a₃ = 10, a₄ = 13 — the sequence is 4, 7, 10, 13, … For the 20th term, no listing needed: a₂₀ = 3·20 + 1 = 61.
Are the sequences 1, 2, 3 and 3, 2, 1 equal? Explain in one sentence.
No. They contain the same numbers but in a different order, and a sequence is an ordered list — so 1, 2, 3 ≠ 3, 2, 1. (As sets they would be equal; as sequences they are not.)
Why is the graph of a sequence a set of separate dots rather than a continuous line?
Because a sequence is a function whose domain is the positive integers ℕ⁺. There is no input between n = 3 and n = 4, so there is no term there and no point to draw — the graph is isolated dots, not a connected curve.
A sequence is given by the recurrence a₁ = 5, aₙ₊₁ = 2aₙ − 1. Find a₂, a₃, a₄.
Apply the step aₙ₊₁ = 2aₙ − 1 repeatedly. a₂ = 2·5 − 1 = 9; a₃ = 2·9 − 1 = 17; a₄ = 2·17 − 1 = 33. The sequence is 5, 9, 17, 33, …
The sequence aₙ = (−1)n·n begins how? Is it increasing, decreasing, or oscillating?
a₁ = (−1)1·1 = −1, a₂ = (−1)2·2 = 2, a₃ = −3, a₄ = 4 — so −1, 2, −3, 4, −5, … The sign flips every step, so it is oscillating (it is neither always rising nor always falling).
The sequence starts 3, 6, 12, … Give two different rules that both produce this start, and say what this shows.
One rule: doubling, aₙ = 3·2n−1 → 3, 6, 12, 24, … Another: the differences 3, 6 themselves growing, e.g. aₙ = the sequence 3, 6, 12, 21, 33, … (add 3, 6, 9, 12, …). Both match 3, 6, 12, then diverge — which shows the first few terms do not determine the rule.
Six questions to lock it in. Tap the answer you think is right.
This lesson opens Stage 26 by framing a sequence as a function with domain the positive integers, aₙ = f(n), so its graph is a set of discrete points — directly supporting CCSS HSF-IF.A.3 ("recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers"). The contrast between explicit general-term formulas and recurrence relations previews HSF-BF.A.2 (writing arithmetic and geometric sequences both recursively and with an explicit formula, and translating between the two). The "first few terms don't fix the rule" caution builds the habit behind MP3 (construct viable arguments) and MP8 (look for and express regularity in repeated reasoning) — a pattern is a conjecture, not yet a proof.