Sequences as Functions (Recursive & Explicit) - Algebra 2

Section 1

Historical Context & Motivation

The study of sequences is one of the oldest pursuits in mathematics. Long before the language of functions was formalized, scholars across cultures noticed that many natural and numerical patterns could be described by simple rules — each term built from the previous one, or each term computed directly from its position. The tension between these two perspectives — recursive (step‐by‐step) and explicit (direct computation) — has shaped mathematics from antiquity to the present day.

c. 200 BCE • Pingala

The Indian mathematician Pingala described what we now call the Fibonacci sequence while analyzing Sanskrit poetic meters. His approach was inherently recursive: each new metrical pattern was constructed from the two preceding ones.

c. 300 BCE • Euclid

Euclid's Elements explored arithmetic progressions (sequences with a constant difference) and geometric progressions (sequences with a constant ratio), establishing explicit formulas for their sums long before the word "function" existed.

1202 • Fibonacci

Leonardo of Pisa (Fibonacci) introduced his famous rabbit‐population sequence to Europe: 1, 1, 2, 3, 5, 8, … He defined it recursively — each term is the sum of the two before it — sparking centuries of investigation into recursive definitions.

1718 • de Moivre & Binet

Abraham de Moivre derived a closed‐form (explicit) formula for the Fibonacci sequence using the golden ratio φ, later refined by Jacques Binet. This showed that even recursively defined sequences can sometimes be expressed as direct formulas — a foundational insight for modern algebra.

1837 • Dirichlet

Peter Gustav Lejeune Dirichlet gave the first rigorous modern definition of a function, paving the way for mathematicians to view sequences as functions whose domain is the set of positive integers (or non‐negative integers). This perspective is the heart of our lesson today.

The central question this lesson addresses is deceptively simple: What is a sequence, and how can we describe it precisely? By treating a sequence as a function — mapping each position number n to its corresponding term — we gain two complementary tools. The explicit formula lets us jump directly to any term, while the recursive formula shows us how the sequence builds upon itself, one step at a time.

Section 2

Core Principles & Definitions

A sequence is an ordered list of numbers called terms. What makes a sequence different from a mere set is that order matters: the first term, the second term, the third term, and so on each have a definite position. Because every term is paired with exactly one position number, a sequence is, by definition, a function whose domain is a subset of the integers (usually {1, 2, 3, …} or {0, 1, 2, …}) and whose range is a subset of the real numbers.

Sequence as a Function

A sequence is a function f : ℕ → ℝ. We write f(n) or, more commonly, a_n to denote the term at position n. The input is the position; the output is the value.

Explicit (Closed‑Form) Rule

An explicit formula computes a_n directly from n alone. You can find the 100th term without knowing the 99th. Example: a_n = 3n + 2.

Recursive Rule

A recursive formula defines each term in relation to one or more previous terms, plus an initial condition (the starting value). Example: a₁ = 5, a_n = a_n−1 + 3.

Domain & Range

Unlike continuous functions, the domain of a sequence is discrete — only whole numbers. This means the graph consists of isolated points, not a continuous curve.

✦

Key Takeaway

Think of an explicit formula as a GPS coordinate — it takes you directly to any term without traveling through the ones before it. A recursive formula is more like driving directions — to reach the 50th term, you follow the route from the 49th, which came from the 48th, all the way back to the starting point. Both describe the same journey; they just navigate differently.

Section 3

Visual Explanation

The diagram below shows how the arithmetic sequence a_n = 2n + 1 can be viewed as a function. The horizontal axis represents the term position n (the domain), while the vertical axis represents the value a_n (the range). Each point is an input‐output pair — just like any function you've graphed — except the points are discrete (no line connecting them) because the domain is the natural numbers.

Notice that the six discrete points lie along what would be a straight line — this is characteristic of an arithmetic sequence, whose explicit formula is linear. The curved violet arrows illustrate the recursive view: to find a₅ = 11, you add 2 to the previous term a₄ = 9. The vertical cyan arrow shows the explicit view: plug n = 5 directly into the formula a_n = 2n + 1 to get 11, bypassing all earlier terms.

Section 4

Mathematical Framework — The Formulas

The two most important families of sequences in Algebra 2 are arithmetic sequences (constant difference between consecutive terms) and geometric sequences (constant ratio between consecutive terms). Each family can be described with both an explicit and a recursive formula.

Arithmetic Sequences

An arithmetic sequence has a common difference d. Starting from a first term a₁, each subsequent term is obtained by adding d. Here are both representations:

Arithmetic — Explicit Formula

aₙ = a₁ + (n − 1) × d

a₁ = first term, d = common difference, n = term position

This formula acts like a direct lookup: for any position n, you compute the value in one step. Notice its structure mirrors a linear function y = mx + b, where the slope corresponds to the common difference d.

Arithmetic — Recursive Formula

a₁ = (given), aₙ = aₙ₋₁ + d for n ≥ 2

Each term equals the previous term plus the common difference

The recursive version emphasizes the process of building the sequence. You must specify the initial condition a₁ — without it, the recursion has no starting point and the sequence is undefined.

Geometric Sequences

A geometric sequence has a common ratio r. Each term is obtained by multiplying the previous term by r:

Geometric — Explicit Formula

aₙ = a₁ × r^(n−1)

a₁ = first term, r = common ratio, n = term position

Geometric — Recursive Formula

a₁ = (given), aₙ = aₙ₋₁ × r for n ≥ 2

Each term equals the previous term times the common ratio

The explicit formula for a geometric sequence is exponential, not linear. This means geometric sequences grow (or decay) much faster than arithmetic ones — a crucial distinction in applications like compound interest, population growth, and radioactive decay.

✦

Key Takeaway

Think of arithmetic sequences as staircases — you climb the same height with every step. Geometric sequences are more like chain reactions — each step multiplies the effect of the one before, creating explosive growth or rapid decay depending on whether the ratio is greater or less than one.

Section 5

Detailed Breakdown — Arithmetic vs. Geometric

Let's compare the two sequence types side by side: their rules, their graphs, and their behavior. The diagram below shows both an arithmetic and a geometric sequence starting from the same first term, making their different growth patterns visually obvious.

Comparison of arithmetic (d = 3) and geometric (r = 2) sequences both starting at a₁ = 2

Both sequences begin at a₁ = 2, but their trajectories diverge dramatically. The arithmetic sequence (d = 3) grows steadily — its points trace a straight line. The geometric sequence (r = 2) starts slowly but explodes upward — its points trace an exponential curve. By the sixth term, the geometric sequence has reached 64 while the arithmetic sequence is only at 17.

Feature	Arithmetic Sequence	Geometric Sequence
Pattern	Add a constant d	Multiply by a constant r
Explicit formula	`aₙ = a₁ + (n−1)d`	`aₙ = a₁ × r^(n−1)`
Recursive formula	`aₙ = aₙ₋₁ + d`	`aₙ = aₙ₋₁ × r`
Graph shape	Linear (straight line of points)	Exponential (curved path of points)
Growth type	Constant (additive)	Multiplicative (compound)
Real-world example	Saving $50/month in a jar	Bacteria doubling every hour

Section 6

Worked Example

Let's work through a complete problem that asks us to move between recursive and explicit representations.

Problem: A sequence is defined recursively as a₁ = 4, aₙ = aₙ₋₁ + 7 for n ≥ 2. Write the explicit formula, find a₂₀, and determine which term equals 200.

1
Step 1 — Identify the Sequence Type

The recursive rule says each term equals the previous term plus 7. Since we add a constant, this is an arithmetic sequence with common difference d = 7 and first term a₁ = 4.
2
Step 2 — Write the Explicit Formula

Substitute into the arithmetic explicit formula: aₙ = 4 + (n − 1) × 7 = 7n − 3. We simplified: 4 + 7n − 7 = 7n − 3.

The explicit formula is aₙ = 7n − 3.
3
Step 3 — Find a₂₀

Plug n = 20 into the explicit formula: a₂₀ = 7(20) − 3 = 140 − 3 = 137. Without the explicit formula, we would have needed to compute all 19 preceding terms using the recursive rule. The explicit formula let us jump directly to the answer.
4
Step 4 — Find Which Term Equals 200

Set aₙ = 200 and solve for n: 200 = 7n − 3 → 203 = 7n → n = 29.

So a₂₉ = 200. The 29th term of the sequence is 200.
5
Step 5 — Verify Consistency

Let's check: a₁ = 7(1) − 3 = 4 ✓, and a₂ = 7(2) − 3 = 11 = 4 + 7 ✓. Our explicit formula is consistent with the original recursive definition.

Section 7

Strengths & Limitations — Recursive vs. Explicit

Both formula types describe the same sequence, but they have different practical strengths and weaknesses. Understanding when to use each is a key part of mathematical fluency.

Criterion	Explicit Formula	Recursive Formula
Finding the nth term	Excellent — direct computation, O(1)	Slow — must compute all preceding terms, O(n)
Finding which term has value v	Excellent — set aₙ = v and solve	Difficult — iterate until you find it
Ease of discovery	Harder — requires recognizing pattern	Easier — often mirrors the verbal description
Modeling real processes	Less intuitive for step-by-step processes	Natural for processes defined by change rules
Computer implementation	Efficient, no memory of past terms needed	Simple to code but may be slow for large n
Existence	Not always possible (e.g., some recursive sequences have no closed form)	Always possible if the rule is defined

✦

Key Takeaway

In practice, you'll often discover a sequence through its recursive pattern ("each term is 3 more than the last") and then convert to an explicit formula for efficiency ("the nth term is 3n + 1"). Mastering the translation between these two views — recursive to explicit and back — is the central skill of this lesson. However, keep in mind that some fascinating sequences (like the Fibonacci sequence or the logistic map) resist easy explicit formulas, making the recursive description their most natural home.

Section 8

Connections to Advanced Theory

The ideas you've learned here form the foundation for several powerful topics you'll encounter later in mathematics. Understanding sequences as functions opens doors to calculus, computer science, and mathematical analysis.

Series and summation. Once you can describe the nth term of a sequence, you can ask: "What happens if I add up all the terms?" This leads to series — the study of infinite sums. The explicit formula for a geometric sequence, for instance, leads directly to the geometric series formula S = a₁ / (1 − r) when |r| < 1, which has applications in physics, finance, and probability.

Recurrence relations. The recursive formulas you've studied are the simplest examples of recurrence relations, a vast area of discrete mathematics. More complex recurrences — like the Fibonacci relation aₙ = aₙ₋₁ + aₙ₋₂ — require techniques from linear algebra (characteristic equations, matrix methods) to solve for explicit formulas.

Limits and convergence. In calculus, you'll study what happens as n → ∞. Does the sequence approach a fixed value (converge) or grow without bound (diverge)? The arithmetic sequence with d ≠ 0 always diverges, while a geometric sequence converges to 0 precisely when |r| < 1.

This Lesson	Advanced Extension
Arithmetic explicit formula	Linear functions, arithmetic series (Gauss's formula)
Geometric explicit formula	Exponential functions, infinite geometric series
Recursive definitions	Recurrence relations, dynamical systems, algorithms
Sequence as a function ℕ → ℝ	Limits of sequences, formal definition of continuity
Domain restricted to integers	Discrete vs. continuous modeling, sampling theory

Section 9

Practice Problems

Question 1: Problem 1 — Conceptual

Explain in your own words why a sequence can be considered a function. What is the domain? What is the range? Why are the points on the graph of a sequence not connected by a line?

Show answer

A sequence is a function because it assigns exactly one output (the term value) to each input (the position number). The domain is the set of positive integers {1, 2, 3, …} (or sometimes starting at 0), and the range is the set of all term values the sequence takes. The points are not connected by a line because the domain is discrete — there is no term at position 1.5 or 2.7, only at whole-number positions. Connecting the dots would imply the existence of values between positions, which is misleading. This contrasts with continuous functions like f(x) = 2x + 1 where every real-number input produces an output.

Question 2: Problem 2 — Basic Identification

Consider the sequence: 5, 10, 20, 40, 80, … Identify whether it is arithmetic or geometric, state the common difference or ratio, write the recursive formula, and write the explicit formula.

Show answer

Check the differences: 10 − 5 = 5, 20 − 10 = 10 — these are not constant, so it's not arithmetic. Check the ratios: 10 ÷ 5 = 2, 20 ÷ 10 = 2, 40 ÷ 20 = 2 — the ratio is constant, so this is a geometric sequence with common ratio r = 2 and first term a₁ = 5. Recursive formula: a₁ = 5, aₙ = aₙ₋₁ × 2 for n ≥ 2. Explicit formula: aₙ = 5 ×

2^{n−1}

. Check: a₁ = 5 × 1 = 5 ✓, a₄ = 5 × 8 = 40 ✓.

Question 3: Problem 3 — Intermediate

An arithmetic sequence has a₃ = 14 and a₇ = 30. Find the common difference d, the first term a₁, the explicit formula, and the value of a₅₀.

Show answer

Using the explicit formula aₙ = a₁ + (n − 1)d, we set up two equations: a₃ = a₁ + 2d = 14 and a₇ = a₁ + 6d = 30. Subtract the first from the second: 4d = 16, so d = 4. Substitute back: a₁ + 2(4) = 14, so a₁ = 14 − 8 = 6. Explicit formula: aₙ = 6 + (n − 1)(4) = 4n + 2. a₅₀ = 4(50) + 2 = 202.

Question 4: Problem 4 — Applied / Multi‑Step

A culture of bacteria starts with 500 organisms and triples every hour. (a) Write both a recursive and an explicit formula for the population after n hours. (b) How many bacteria are present after 6 hours? (c) After how many complete hours will the population first exceed 1,000,000?

Show answer

(a) This is a geometric sequence with a₀ = 500 and r = 3 (using n = 0 for the start). Recursive: a₀ = 500, aₙ = 3 × aₙ₋₁ for n ≥ 1. Explicit: aₙ = 500 × 3ⁿ. (b) a₆ = 500 × 3⁶ = 500 × 729 = 364,500 bacteria. (c) Solve 500 × 3ⁿ > 1,000,000. Divide: 3ⁿ > 2,000. Take logarithms: n > log(2000) / log(3) ≈ 3.301 / 0.477 ≈ 6.92. Since n must be a whole number and a₆ = 364,500 < 1,000,000, we need n = 7. Check: a₇ = 500 × 2187 = 1,093,500 > 1,000,000 ✓.

Question 5: Problem 5 — Critical Thinking / Synthesis

A sequence is defined recursively as: a₁ = 2, a₂ = 6, aₙ = aₙ₋₁ + aₙ₋₂ for n ≥ 3. (a) Write out the first eight terms. (b) Is this sequence arithmetic, geometric, or neither? Justify your answer. (c) Can you write a simple explicit formula for this sequence? Why or why not? (d) How does this sequence relate to the Fibonacci sequence?

Show answer

(a) a₁ = 2, a₂ = 6, a₃ = 8, a₄ = 14, a₅ = 22, a₆ = 36, a₇ = 58, a₈ = 94. (b) Neither. The differences are 4, 2, 6, 8, 14, 22, 36 — not constant (so not arithmetic). The ratios are 3, 4/3, 7/4, 11/7, 18/11, 29/18, 47/29 — not constant (so not geometric). (c) A "simple" algebraic explicit formula in the way arithmetic (linear) or geometric (exponential) sequences have is not possible using elementary operations. This is a second-order linear recurrence, similar to Fibonacci. An explicit formula does exist via the characteristic equation method (involving irrational roots of x² − x − 1 = 0), but it's far from "simple." This illustrates a key limitation: not all recursive sequences can be easily converted to explicit form. (d) This sequence follows the exact same recursive rule as the Fibonacci sequence — each term is the sum of the two preceding terms — but with different initial conditions. The Fibonacci sequence starts F₁ = 1, F₂ = 1, while ours starts 2, 6. In fact, this sequence can be expressed as aₙ = 2Fₙ₋₂ + 6Fₙ₋₁ for appropriate Fibonacci indexing, showing how initial conditions reshape the entire sequence even when the recursive rule is identical.

Summary

Lesson Summary

Summary

A sequence is a function whose domain is the positive integers, mapping each position n to a term value a_n. Every sequence can be described in two complementary ways: an explicit (closed‑form) formula that computes any term directly from its position, and a recursive formula that builds each term from one or more previous terms plus an initial condition. For arithmetic sequences, the explicit formula is linear — a_n = a₁ + (n − 1)d — reflecting constant additive growth, while the recursive rule simply adds the common difference d at each step. For geometric sequences, the explicit formula is exponential — a_n = a₁ × rⁿ⁻¹ — reflecting multiplicative growth governed by the common ratio r.

The explicit formula excels at direct computation and solving for unknown positions, while the recursive formula excels at modeling step‑by‑step processes and is often easier to discover from a verbal description. Some sequences — like the Fibonacci sequence — have no simple explicit form, reminding us that the recursive perspective is sometimes the more fundamental one. Mastering the translation between recursive and explicit representations is a cornerstone skill that connects algebra to calculus, computer science, and mathematical modeling.