Why the racer stuck far behind is the one nobody can catch later.
Someone saving $1,000 a month vs. $100 growing 10% a month — who reaches $100,000 first?
Grow by adding or by multiplying — the sequence decides the future.
Experiment
Hands-on experiment
Both start at 100 — left adds +100, right multiplies ×1.5
| Arithmetic (+100) | Geometric (×1.5) | |
|---|---|---|
| term 1 | 100 | 100 |
Read more — why it exists · insights · common mistakes · formulasExpand ▾
Why
Why does this exist?
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.
Insight
Insights from the video
“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.
Misconception
Common misconceptions
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.
Formula
Writing it as math
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.
In Real Life
Where you meet it in real life
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.
Connection
Concepts connect
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 →