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Lesson 16 — Numerical Sequences

Sequence as a function with domain ℕ. Recurrence relations, monotonicity, boundedness. Gateway to limits.

Used in: 1st year of HS (age 15) · Math B Japanese (ch. 数列) · Calculus I — US — preview

(a_n)_{n \in \mathbb{N}}, \quad a_n = f(n)

A sequence is a function from $\mathbb{N}$ to $\mathbb{R}$ . Each $n$ receives a term $a_n$ . Sequences are central objects in analysis (limit, series, convergence) and the foundation of formal calculus — they prepare Lesson 19 (preview) and Lesson 41 (formal limit).

Choose your door

Rigorous notation, full derivation, hypotheses

Definition and properties

How to describe a sequence

Explicit formula (general term): $a_n = 2n + 1$ — terms $3, 5, 7, 9, \ldots$
Recurrence: $a_1 = 1$ , $a_{n+1} = a_n + 2$ — same result.
Description: "the n-th prime number" — $2, 3, 5, 7, 11, 13, \ldots$ (no closed form).

Monotonicity

Increasing: $a_{n+1} > a_n \quad \forall n$ .
Non-decreasing: $a_{n+1} \geq a_n$ .
Decreasing: $a_{n+1} < a_n$ .
Constant: $a_{n+1} = a_n$ .

Boundedness

$(a_n)$ is bounded if there exists $M > 0$ with $|a_n| \leq M$ for all $n$ . Bounded above if $a_n \leq M_+$ ; bounded below if $a_n \geq M_-$ .

Intuitive convergence (formalized in Lesson 41)

$(a_n)$ converges to $L$ if " $a_n$ gets arbitrarily close to $L$ as $n$ becomes large". Formally: $\lim_{n \to \infty} a_n = L \iff \forall \varepsilon > 0,\ \exists N : n \geq N \Rightarrow |a_n - L| < \varepsilon$

Famous sequences

Name	Definition	Terms
Natural numbers	$a_n = n$	$1, 2, 3, \ldots$
Squares	$a_n = n^2$	$1, 4, 9, 16, \ldots$
Harmonic	$a_n = 1/n$	$1, 1/2, 1/3, \ldots$
Fibonacci	$F_{n+2} = F_{n+1} + F_n$ , $F_1 = F_2 = 1$	$1, 1, 2, 3, 5, 8, 13, \ldots$
Geometric	$a_n = q^n$	$q, q^2, q^3, \ldots$

"A sequence is just a list of numbers, but in Math 2E we make this list infinite." — Active Calculus §8.2

Worked examples

Example— 1· List terms from an explicit formula

Problem: List the first 5 terms of the sequence $a_n = 2n + 1$ .

Strategy: Substitute $n = 1, 2, 3, 4, 5$ and calculate each $a_n$ .

Solution:

$a_1 = 2(1) + 1 = 3$ .
$a_2 = 2(2) + 1 = 5$ .
$a_3 = 2(3) + 1 = 7$ .
$a_4 = 2(4) + 1 = 9$ .
$a_5 = 2(5) + 1 = 11$ .

$a_1, a_2, a_3, a_4, a_5 = 3, 5, 7, 9, 11.$

Verification: The difference between consecutive terms is constant $= 2$ — characteristic of an arithmetic progression with $a_1 = 3$ and common difference $d = 2$ . Agrees with $a_n = a_1 + (n-1)d = 3 + 2(n-1) = 2n + 1$ . Checks out.

Source. Active Calculus §8.2 — Boelkins · CC-BY-NC-SA.

Example— 2· Calculate terms from a recurrence

Problem: Calculate the first 6 terms of the Fibonacci sequence, defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} + F_n$ .

Strategy: Apply the recurrence repeatedly, adding the last two terms to get the next.

Solution:

$F_1 = 1$ (given).
$F_2 = 1$ (given).
$F_3 = F_2 + F_1 = 1 + 1 = 2$ .
$F_4 = F_3 + F_2 = 2 + 1 = 3$ .
$F_5 = F_4 + F_3 = 3 + 2 = 5$ .
$F_6 = F_5 + F_4 = 5 + 3 = 8$ .

$F_1, \ldots, F_6 = 1, 1, 2, 3, 5, 8.$

Verification: The ratio $F_{n+1}/F_n$ approaches the golden ratio $\phi = (1 + \sqrt{5})/2 \approx 1.618$ . $F_6/F_5 = 8/5 = 1.6$ . On track.

Source. Hammack — Book of Proof §10.4 — free license.

Example— 3· Find the explicit formula from terms

Problem: Find the general term of the sequence $\tfrac{1}{2}, \tfrac{1}{4}, \tfrac{1}{8}, \tfrac{1}{16}, \tfrac{1}{32}, \ldots$

Strategy: Identify the pattern. Each term is half the previous — geometric sequence with ratio $1/2$ . The term $a_n$ should have $2^n$ in the denominator.

Solution:

Observe the pattern. $a_1 = 1/2 = 1/2^1$ . $a_2 = 1/4 = 1/2^2$ . $a_3 = 1/8 = 1/2^3$ .
Conjecture. $a_n = 1/2^n$ .
Verify. $a_4 = 1/2^4 = 1/16$ . $a_5 = 1/2^5 = 1/32$ . Both check.

$a_n = \frac{1}{2^n}.$

Verification: Ratio $a_{n+1}/a_n = (1/2^{n+1})/(1/2^n) = 1/2$ . Confirms the observed ratio.

Source. Lebl — Basic Analysis Vol. I §2.1 — CC-BY-SA license.

Example— 4· Monotonicity and boundedness

Problem: Show that the sequence $a_n = \dfrac{n+1}{n}$ is decreasing and bounded below by 1.

Strategy: To show decrease, demonstrate $a_{n+1} - a_n < 0$ . To show lower bound by 1, demonstrate $a_n > 1$ for all $n$ .

Solution:

Rewrite. $a_n = 1 + 1/n$ .
Show it is decreasing. $a_{n+1} - a_n = (1 + 1/(n+1)) - (1 + 1/n) = 1/(n+1) - 1/n = (n - (n+1))/(n(n+1)) = -1/(n(n+1)) < 0$ . So $a_{n+1} < a_n$ for all $n \geq 1$ .
Show lower bound. Since $1/n > 0$ for all $n \geq 1$ , $a_n = 1 + 1/n > 1$ . Thus $a_n > 1$ is a lower bound.
The sequence is also bounded above by $a_1 = 2$ (the first term, the maximum of a decreasing sequence).

$a_n \in (1, 2]\ \text{for all}\ n \geq 1,\ \text{and decreasing.}$

Verification: $a_1 = 2$ , $a_2 = 3/2 = 1.5$ , $a_3 = 4/3 \approx 1.333$ , $a_{10} = 1.1$ , $a_{1000} = 1.001$ . Each term is smaller than the previous and greater than 1. As $n$ grows, $a_n \to 1$ . Matches the theory.

Source. Lebl — Basic Analysis Vol. I §2.1 — CC-BY-SA license.

Example— 5· Coffee cooling as a recurrence (real modeling)

Problem: The temperature of cooling coffee follows $T_n = 65 \cdot (0.9)^n + 25$ , where $n$ is time in minutes and the initial temperature is $T_0 = 65 + 25 = 90$ °C. What value does $T_n$ approach as $n \to \infty$ ?

Strategy: The part $(0.9)^n$ is a geometric sequence with ratio between 0 and 1, so it approaches 0. The limit of the entire sequence is the constant term.

Solution:

Identify the structure. $T_n = A \cdot q^n + B$ with $A = 65$ , $q = 0.9$ , $B = 25$ .
Behavior of $q^n$ . Since $|q| = 0.9 < 1$ , $q^n \to 0$ as $n \to \infty$ .
Substitute intuitively. $T_n \to 65 \cdot 0 + 25 = 25$ °C.
Physical interpretation. The term $25$ °C is the ambient temperature. The coffee cools asymptotically to that temperature (Newton's law of cooling in discrete form).

$\lim_{n \to \infty} T_n = 25\ °\text{C}.$

Verification: $T_0 = 65 + 25 = 90$ °C. $T_5 = 65 \cdot (0.9)^5 + 25 \approx 65 \cdot 0.59 + 25 \approx 63.4$ °C — consistent with lukewarm coffee after 5 minutes. The limit remains $25$ °C.

Source. Active Calculus §8.2 — Boelkins · CC-BY-NC-SA.

Exercise list

40 exercises · 10 with worked solution (25%)

Application 21Understanding 12Modeling 4Challenge 2Proof 1

Sources

OpenStax Algebra and Trigonometry 2e — Jay Abramson et al. · 2022, 2nd ed · EN · CC-BY 4.0 · §9.1: sequences, notation, explicit formulas, recurrences, summation.
Stitz-Zeager Precalculus — Carl Stitz, Jeff Zeager · 2013 · EN · CC-BY-NC-SA · §9.1: sequences, monotonicity, boundedness, patterns.
Basic Analysis: Introduction to Real Analysis (Vol. I) — Jiří Lebl · 2024, v6.0 · EN · CC-BY-SA · §2.1: sequences, monotonicity, boundedness, convergence.
Active Calculus — Matt Boelkins · 2024 · EN · CC-BY-NC-SA · §8.2: sequences and intuitive convergence.