Sequences

Why this matters

In programming, you often handle arrays, lists, streams, and time series. In mathematics, a sequence is the clean, precise version of that idea: an ordered list where order is not optional—it is the point.

A big payoff of learning sequences early is that they become the common language for later topics:

• •Recurrence relations define aₙ in terms of earlier terms.
• •Generating functions package all aₙ into one object.
• •Taylor series are sequences of coefficients that approximate functions.

Definition (with intuition)

A sequence is a function whose input is an index and whose output is a number.

Most commonly, indices come from the positive integers:

• •n ∈ {1, 2, 3, …}

We write the value at index n as aₙ (read “a sub n”).

So you can think of a sequence as:

• •a: ℕ⁺ → ℝ (or → ℤ, or → ℂ, depending on the context)
• •n ↦ aₙ

Ordered list viewpoint

You’ll also see sequences written as an ordered list:

• •(a₁, a₂, a₃, …)

The parentheses emphasize order: (1, 2) ≠ (2, 1).

Indexing conventions (don’t get tripped up)

Different fields start counting at different places:

• •In many math texts: start at n = 1.
• •In computer science: arrays often start at index 0.

Both are valid. What matters is that you declare your indexing.

Example:

• •If aₙ = 2n, then a₁ = 2, a₂ = 4, a₃ = 6, …
• •If you instead start at n = 0, then a₀ = 0, a₁ = 2, a₂ = 4, …

A sequence is not necessarily “patterned”

Many sequences follow a simple rule, but a sequence can be any mapping from indices to values. For instance:

• •aₙ = 0 if n is even, and aₙ = 1 if n is odd
• •aₙ = the n-th digit of π

This lesson focuses on two foundational “patterned” types:

• •Arithmetic sequences: constant additive change
• •Geometric sequences: constant multiplicative change

Quick vocabulary

This vocabulary will matter when you later learn recurrence relations and series.

Why arithmetic sequences are a “first pattern”

If you watch how something changes over time and the change is constant, you are in arithmetic-sequence territory.

Examples in the real world:

• •Saving a fixed amount of money each month
• •Moving at constant velocity: position increases by a constant distance per unit time
• •A linear function sampled at equally spaced inputs

The defining idea is: each step adds the same amount.

Definition

A sequence (aₙ) is arithmetic if the difference between consecutive terms is constant:

• •aₙ − aₙ₋₁ = d for all n ≥ 2

Here d is the common difference.

Building intuition with a small table

Suppose a₁ = 5 and d = 3.

Each time n increases by 1, the value increases by 3.

Explicit formula (the “closed form”)

If you know the first term a₁ and the common difference d, you can compute any term without listing all previous ones.

Start from the pattern:

• •a₂ = a₁ + d
• •a₃ = a₁ + 2d
• •a₄ = a₁ + 3d

So in general:

• •aₙ = a₁ + (n − 1)d

This is worth deriving carefully because it’s the template for many later topics.

Derivation (showing the work):

We are adding d repeatedly:

• •from a₁ to a₂: add d once
• •from a₁ to a₃: add d twice
• •from a₁ to aₙ: add d (n − 1) times

So:

• •aₙ = a₁ + d + d + … + d (with (n − 1) copies of d)
• •aₙ = a₁ + (n − 1)d

Detecting an arithmetic sequence from data

Given terms a₁, a₂, a₃, … compute consecutive differences:

• •Δₙ = aₙ − aₙ₋₁

If all Δₙ are equal, the sequence is arithmetic.

Example:

• •(2, 7, 12, 17, …)

Differences: 5, 5, 5, … ⇒ arithmetic with d = 5.

Arithmetic sequences connect to linear functions

If aₙ = a₁ + (n − 1)d, that’s linear in n:

• •aₙ = dn + (a₁ − d)

So arithmetic sequences are essentially “linear growth” indexed by integers.

Sum of the first n terms (a very common task)

You will frequently want:

• •Sₙ = a₁ + a₂ + … + aₙ

For an arithmetic sequence, there is a famous formula:

• •Sₙ = n(a₁ + aₙ)/2

Why this is true (pairing idea):

Write the sum forward and backward:

Sₙ = a₁ + a₂ + … + aₙ₋₁ + aₙ

Sₙ = aₙ + aₙ₋₁ + … + a₂ + a₁

Add them termwise:

2Sₙ = (a₁ + aₙ) + (a₂ + aₙ₋₁) + … + (aₙ + a₁)

Each pair equals (a₁ + aₙ), and there are n pairs:

2Sₙ = n(a₁ + aₙ)

Divide by 2:

Sₙ = n(a₁ + aₙ)/2

You can also substitute aₙ = a₁ + (n − 1)d if you want Sₙ in terms of a₁, d, n.

When arithmetic sequences fail

Not every “nice-looking” sequence is arithmetic.

Example:

• •(1, 2, 4, 7, 11, …)

Differences: 1, 2, 3, 4, … not constant ⇒ not arithmetic.

This is a key habit: check the differences rather than guessing.

Why geometric sequences are the “second pattern”

If something changes by a constant percentage (or constant factor), you get multiplicative growth or decay.

Examples:

• •Compound interest (multiply by 1.05 each period for 5% growth)
• •Population growth models (simplified)
• •Repeated halving/doubling in algorithms
• •Exponential functions sampled at equally spaced inputs

The defining idea is: each step multiplies by the same factor.

Definition

A sequence (aₙ) is geometric if the ratio between consecutive terms is constant:

• •aₙ / aₙ₋₁ = r for all n ≥ 2 (assuming aₙ₋₁ ≠ 0)

Here r is the common ratio.

Example table

Suppose a₁ = 3 and r = 2.

Each step doubles.

Explicit formula

From the pattern:

• •a₂ = a₁r
• •a₃ = a₁r²
• •a₄ = a₁r³

So in general:

• •aₙ = a₁ rⁿ⁻¹

Derivation (showing the work):

You multiply by r repeatedly:

• •from a₁ to a₂: multiply by r once
• •from a₁ to a₃: multiply by r twice
• •from a₁ to aₙ: multiply by r (n − 1) times

So:

• •aₙ = a₁ · r · r · … · r (with (n − 1) copies of r)
• •aₙ = a₁ rⁿ⁻¹

Detecting a geometric sequence

Compute ratios:

• •ρₙ = aₙ / aₙ₋₁

If all ρₙ are equal (and denominators aren’t 0), it’s geometric.

Example:

• •(81, 27, 9, 3, …)

Ratios: 1/3, 1/3, 1/3, … ⇒ geometric with r = 1/3.

Signs and special cases

• •If r is negative, the sequence alternates signs.

Example: a₁ = 2, r = −3 ⇒ (2, −6, 18, −54, …)

• •If 0 < r < 1, the terms shrink toward 0.
• •If r = 1, the sequence is constant: aₙ = a₁.
• •If r = 0, then a₂ = 0 and all later terms are 0.

Sum of the first n terms

Let:

• •Sₙ = a₁ + a₂ + … + aₙ

For geometric sequences (r ≠ 1):

• •Sₙ = a₁(1 − rⁿ)/(1 − r)

Derivation (showing the work):

Start with:

Sₙ = a₁ + a₁r + a₁r² + … + a₁rⁿ⁻¹

Multiply both sides by r:

rSₙ = a₁r + a₁r² + … + a₁rⁿ

Subtract:

Sₙ − rSₙ = (a₁ + a₁r + … + a₁rⁿ⁻¹) − (a₁r + … + a₁rⁿ)

Everything cancels except the first term a₁ and the last term −a₁rⁿ:

(1 − r)Sₙ = a₁ − a₁rⁿ

Factor:

(1 − r)Sₙ = a₁(1 − rⁿ)

Divide:

Sₙ = a₁(1 − rⁿ)/(1 − r)

(If r = 1, then Sₙ = n a₁.)

Infinite geometric series (a preview)

If |r| < 1, then rⁿ → 0 as n → ∞, and the sums approach a finite limit:

• •a₁ + a₁r + a₁r² + … = a₁/(1 − r)

You’ll revisit this idea later in generating functions and Taylor series.

Two ways to define a sequence

Sequences are often described in one of two styles:

1) Explicit: aₙ is given directly in terms of n.

• •Example (arithmetic): aₙ = a₁ + (n − 1)d
• •Example (geometric): aₙ = a₁ rⁿ⁻¹

2) Recursive: aₙ is given in terms of previous terms.

• •Example (arithmetic): a₁ = 5, and aₙ = aₙ₋₁ + 3 for n ≥ 2
• •Example (geometric): a₁ = 3, and aₙ = 2aₙ₋₁ for n ≥ 2

Why recursion matters later

Recursive definitions are natural in algorithms (“do the next step from the previous step”) and in discrete math (“build the next object from smaller ones”). This is the doorway to:

• •Recurrence Relations: more complex recursion like aₙ = 3aₙ₋₁ − 2aₙ₋₂
• •Generating Functions: turning (a₀, a₁, a₂, …) into a power series A(x) = ∑ aₙ xⁿ
• •Taylor Series: coefficients (a₀, a₁, a₂, …) approximate a function locally

A unifying viewpoint: “constant change”

Arithmetic and geometric sequences are special because their “change rule” is constant:

This isn’t just classification—it helps you choose tools.

• •Constant differences ⇒ look for linear formulas.
• •Constant ratios ⇒ look for exponential formulas.

Mini-connection to functions and sampling

If you sample a function at integer points, you get a sequence:

• •Let f(n) be a function; then define aₙ = f(n).

Examples:

• •If f(x) = 2x + 1, then aₙ = 2n + 1 is arithmetic (difference 2).
• •If f(x) = 3·2ˣ, then aₙ = 3·2ⁿ is geometric (ratio 2).

Mini-connection to vectors (notation practice)

Later, you’ll often package multiple sequences into a vector sequence, like xₙ ∈ ℝᵏ.

For example, a 2D position over time:

• •xₙ = (xₙ, yₙ)

Even here, the same patterns can apply:

• •arithmetic-like: xₙ = x₁ + (n − 1)d (constant step vector d)
• •geometric-like scaling: xₙ = rⁿ⁻¹ x₁

You won’t need this vector form yet, but it shows how sequences generalize cleanly.

What you should be able to do after this node

• •Interpret aₙ as “the value at index n”
• •Identify arithmetic sequences by constant differences
• •Identify geometric sequences by constant ratios
• •Write explicit formulas from a₁ and d or r
• •Compute partial sums Sₙ for arithmetic and geometric sequences

Those abilities are exactly the prerequisites for recurrence relations, generating functions, and series.