Sequences

Drop a bouncy ball from a height of 2 metres. Each bounce reaches 60\% of the height before it, so the peak heights (in metres) run

2,\ 1.2,\ 0.72,\ 0.432,\ 0.2592,\ \dots

Or open a savings account with £100 and add £100 every month: the balance reads 100, 200, 300, 400, \dots Or press \sqrt{\phantom{x}} on a calculator over and over starting from 10: 10,\ 3.162,\ 1.778,\ 1.334,\ 1.155,\ \dots Three very different situations, one common shape: an ordered, never-ending list of numbers produced by a rule. The bounce heights are marching down toward 0; the square roots are creeping toward 1; the balance climbs forever. Where is the list heading? is the question this whole chapter turns on — and the list itself is called a sequence.

a_1,\ a_2,\ a_3,\ a_4,\ \dots

The clean way to say it: a sequence is just a function whose inputs are the natural numbers. Feed it a position n \in \{1, 2, 3, \dots\} and it hands back the n-th term, a_n:

a : \mathbb{N} \to \mathbb{R}, \qquad n \mapsto a_n.

We write the whole sequence as (a_n)_{n \ge 1}, or just (a_n). The subscript is the input; the term is the output. So "the sequence of squares" 1, 4, 9, 16, \dots is the rule a_n = n^2 — a single formula, the closed form, that produces any term you ask for without working up to it. The bouncing ball above is a_n = 2 \cdot (0.6)^{\,n-1}; the savings account is a_n = 100n.

Spot the pattern, write the closed form

Most of the work of a first course on sequences is reading a list and recovering its rule. Here are the four patterns you will meet again and again. In each, watch the same move: stare at consecutive terms, find what stays constant, and turn it into a formula for a_n.

Step 1 — Arithmetic: a constant difference. Take 3, 7, 11, 15, \dots Each term is 4 more than the last, so the gap d = a_{n+1} - a_n = 4 never changes. Starting at a_1 = 3 and adding 4 a total of n-1 times:

a_n = a_1 + (n-1)\,d = 3 + 4(n-1) = 4n - 1.

Sanity-check the formula against a term you didn't use to build it: a_4 = 4\cdot4 - 1 = 15. It matches the list — always do this check when you reverse-engineer a rule from terms.

Step 2 — Geometric: a constant ratio. Take 2, 6, 18, 54, \dots Now each term is 3 times the last, so the ratio r = a_{n+1}/a_n = 3 is fixed. Starting at a_1 = 2 and multiplying by 3 a total of n-1 times:

a_n = a_1\, r^{\,n-1} = 2 \cdot 3^{\,n-1}.

Step 3 — Harmonic: the reciprocals of the counting numbers. The list 1, \tfrac12, \tfrac13, \tfrac14, \dots has neither a constant difference nor a constant ratio, but its closed form is the simplest of all:

a_n = \frac{1}{n}.

Its terms shrink toward 0 — a fact we will pin down exactly when we define the limit of a sequence.

Step 4 — Recursive: a term defined from earlier terms. Sometimes the rule is not a closed form at all, but a recurrence — a starting value plus an instruction for the next term. The Fibonacci sequence is the classic:

a_1 = 1,\quad a_2 = 1,\qquad a_{n} = a_{n-1} + a_{n-2}\ \ (n \ge 3),

giving 1, 1, 2, 3, 5, 8, 13, \dots A recurrence always tells you how to grind out the next term; whether a tidy closed form also exists is a separate, often harder, question.

A sequence of real numbers is a function

a : \mathbb{N} \to \mathbb{R}, \qquad a(n) = a_n,

written (a_n)_{n \ge 1}. It may be specified two ways, which are interchangeable when both exist:

Nothing forces a sequence to start at n = 1. Power series and computer scientists love starting at n = 0, so (a_n)_{n \ge 0} runs a_0, a_1, a_2, \dots Either convention is fine — it only shifts the bookkeeping. A geometric sequence indexed from 0 reads a_n = a_0\, r^{\,n} (no n-1), which is why the geometric series is usually written \sum_{n \ge 0} a r^n. Pick a convention, state it, and stay consistent.

See the terms march out

A sequence lives most naturally as dots above the positions 1, 2, 3, \dots on the horizontal axis — the height of each dot is the term a_n. There is no curve joining them: the inputs are whole numbers, so the graph of a sequence is a scatter of isolated points, one per position. That scatter is where a sequence's personality shows.

Switch the rule below and watch the shape change. The arithmetic rule 4n - 1 climbs in a dead-straight staircase — same step up every time, off to infinity. The geometric rule 2 \cdot 3^{\,n-1} curves away so violently it escapes the frame within a handful of terms. The harmonic rule 1/n does the opposite: it sinks toward the axis, each dot squeezed closer to height 0 but never touching it. And Fibonacci, the recursive rule, starts sleepily — 1, 1, 2, 3 — then rockets off even faster than you expect. Same picture, four utterly different destinies. Which sequences settle down and which run away is exactly the question we take up next.

Worked example: terms that creep toward 1

Take the rule a_n = \dfrac{n}{n+1} and simply write out terms:

a_1 = \tfrac12, \quad a_2 = \tfrac23, \quad a_3 = \tfrac34, \quad a_4 = \tfrac45, \quad a_{10} = \tfrac{10}{11} \approx 0.909, a_{100} = \tfrac{100}{101} \approx 0.990, \qquad a_{10\,000} = \tfrac{10\,000}{10\,001} \approx 0.9999.

The terms are plainly creeping up toward 1 — and never reaching it, since the numerator is always one short of the denominator. To see how fast, rewrite the rule:

a_n = \frac{n}{n+1} = \frac{(n+1) - 1}{n+1} = 1 - \frac{1}{n+1}.

So the gap between a_n and 1 is exactly \tfrac{1}{n+1}, and that gap shrinks below any tolerance you name: below 0.01 once n \ge 100, below 0.000001 once n \ge 10^6. We say the sequence converges to 1, and call 1 its limit:

\lim_{n \to \infty} \frac{n}{n+1} = 1, \qquad \text{or} \qquad a_n \to 1.

Here is the picture. Choose a tolerance \varepsilon with the slider: it draws a band from 1 - \varepsilon to 1 + \varepsilon around the limit. However thin you make the band, there is always a cut-off position — the dashed vertical line — beyond which every single later term sits inside the band. That "for any tolerance, eventually all terms are inside" is precisely what "approaches" means, and it becomes the formal definition of a limit.

Worked example: a recursive sequence finds its level

Closed forms let you jump straight to a_{1000}; recurrences make you walk. Try this one:

a_1 = 1, \qquad a_{n+1} = \frac{a_n}{2} + 1.

Halve the current term, add 1, repeat. Grinding out terms:

a_1 = 1,\quad a_2 = 1.5,\quad a_3 = 1.75,\quad a_4 = 1.875,\quad a_5 = 1.9375,\quad a_6 = 1.96875.

The terms climb, but by less each time — the increments are 0.5, 0.25, 0.125, 0.0625, \dots, halving at every step. It looks for all the world as if the sequence is homing in on 2. Can we do better than "it looks like"? Yes — with a beautiful trick. Suppose the sequence does settle down to some limit L. Then for enormous n, both a_n and a_{n+1} are essentially L, so the recurrence itself must hold with L in both slots:

L = \frac{L}{2} + 1 \quad\Longrightarrow\quad \frac{L}{2} = 1 \quad\Longrightarrow\quad L = 2.

The limit, if it exists, can only be 2 — the fixed point of the rule, the one value the rule maps to itself (\tfrac{2}{2} + 1 = 2). And here it genuinely does exist: the gap to 2 is 1, 0.5, 0.25, 0.125, \dots — each step exactly halves the remaining distance, so the gap is (\tfrac12)^{\,n-1} and dies away to 0. Mind the italics, though: the fixed-point equation only names the candidate. The rule a_{n+1} = 2a_n has fixed point 0, yet from a_1 = 1 it explodes to infinity — the candidate is only the limit when the sequence actually converges.

Fibonacci introduced his sequence in 1202 with a puzzle about immortal breeding rabbits — each month's pair count is the sum of the previous two months'. The sequence itself blows up to infinity, so it has no limit. But look at the ratios of consecutive terms:

\tfrac{1}{1} = 1,\quad \tfrac{2}{1} = 2,\quad \tfrac{3}{2} = 1.5,\quad \tfrac{5}{3} \approx 1.667,\quad \tfrac{8}{5} = 1.6,\quad \tfrac{13}{8} = 1.625,\quad \tfrac{21}{13} \approx 1.615,\ \dots

The ratios bounce alternately above and below, closing in on \varphi = \tfrac{1 + \sqrt{5}}{2} \approx 1.618 — the golden ratio. The fixed-point trick explains why: if r_n = a_{n+1}/a_n settles to L, the recurrence forces L = 1 + \tfrac{1}{L}, i.e. L^2 = L + 1, whose positive root is exactly \varphi. A divergent sequence secretly manufacturing one of the most famous constants in mathematics — sequences built from other sequences is a move you will see constantly from here on.

Three destinies: converge, diverge, oscillate

Every sequence you meet does one of three things in the long run — and classifying the behaviour is usually the very first thing to do.

Note the asymmetry: "converges" is one very specific fate, and everything else is called divergent — whether the terms explode, plummet, or just dither. A bounded sequence can still diverge, as (-1)^n shows: staying trapped between -1 and 1 is not the same as settling on one value.

Three classic traps, each of which poisons the rest of the chapter if it takes hold:

Here is a recurrence a child can run: start from any positive whole number; if the current term is even, halve it; if odd, triple it and add 1. From 7:

7,\ 22,\ 11,\ 34,\ 17,\ 52,\ 26,\ 13,\ 40,\ 20,\ 10,\ 5,\ 16,\ 8,\ 4,\ 2,\ 1.

Every starting number ever tried — beyond 2^{68} by computer — crashes down to 1 in the end. Does every starting number? That is the Collatz conjecture, open since 1937. Erdős said of it, "Mathematics is not yet ripe for such questions." A one-line rule about a sequence, and the entire mathematical world cannot decide where the list is heading. Long-run behaviour of sequences is a genuinely deep question — which is why this chapter builds the tools carefully.

See it explained