aₙ · Sequence
Grow by adding or by multiplying — the sequence decides the future.
Someone saving $1,000 a month vs. $100 growing 10% a month — who reaches $100,000 first?
Why the racer stuck far behind is the one nobody can catch later.
Open the immersive version →Why does this question matter?
Things that unfold over time — monthly savings, yearly interest, infections per generation — all share the structure 'the next term is determined by the previous one'.
If the rule is 'add a fixed amount', it's an arithmetic sequence; if it's 'multiply by a fixed ratio', it's geometric. The first makes a steady straight line; the second starts sluggish, then explodes.
The fact that the geometric racer always eventually overtakes — that's the backbone for understanding compound investing, technological progress, and epidemic spread. Sequences are the first tool that makes time computable.
Pause and try it yourself
Make a prediction first. Change one input at a time and compare what happens with what you expected.
Both start at 100 — left adds +100, right multiplies ×1.5
| Arithmetic (+100) | Geometric (×1.5) | |
|---|---|---|
| term 1 | 100 | 100 |
Read the idea from another angle
A sequence is a technique for turning time into numbers.
If a function is 'input→output', a sequence is a function whose inputs are 1st, 2nd, 3rd… That's why every phenomenon with a time order — savings, interest, population — gets modeled as a sequence.
A recurrence is the mindset of 'defining today by yesterday'.
Determining the next term from the previous one, like aₙ₊₁ = aₙ × 1.1, is a recurrence relation. Loan balances, recursion in programming, even fractals — it's the language of repeating structure.
Now read the definitions with their conditions
Let's write the two growth rules from the race as formulas.
Arithmetic sequence — additive growth
The first term a₁ with the common difference d added (n−1) times. Plotted, it's a straight line (sibling of the linear function).
Geometric sequence — multiplicative growth
The first term multiplied by the common ratio r, (n−1) times. Plotted, it's an exponential curve (sibling of the exponential function).
Recurrence relations
'Next = now + d' or 'next = now × r'. A loan balance combines both: balanceₙ₊₁ = balanceₙ × (1+rate) − payment.
Make one more distinction
Whoever leads at the start keeps leading.
Arithmetic wins the early game; geometric wins the endgame. Multiplicative growth scales with its own size — slow while small, but once past a certain size, nothing catches it.
Sequence formulas (aₙ = a + (n−1)d) are separate knowledge to memorize.
The formula is just 'start from the first term and add d, (n−1) times' written down verbatim. If you can say the rule in words, the formula writes itself.
Where else can this idea help?
Savings vs investing
Fixed monthly deposits grow arithmetically; compound investing with reinvested returns grows geometrically. The younger you are, the longer the geometric clock runs — the basis of 'start early'.
Loan repayment schedules
The balance calculation in equal-installment loans (see the Loan lab) is exactly a recurrence. The monthly payment formula comes from the sum of a geometric sequence.
The pace of technology
Chip density doubling every two years (Moore's law) is a geometric sequence. It's why technological progress feels like it 'suddenly' speeds up.
Recursion in programming
Self-referential definitions like the Fibonacci sequence (aₙ = aₙ₋₁ + aₙ₋₂) are the prototype of recursive algorithms.
Explain it in your own words
What question comes next?
Previous concept
Functions
A sequence is a function whose input is 'which position?' (a natural number). Seen through function eyes, sequence formulas become natural.
← Functions labLeads to next
Limits
If a sequence goes on forever, where does it arrive? The moment you ask about that destination, limits — the door to calculus — begin.
Go to the Limits lab →