11.1 · Sequences and Their Notations
Explore ordered lists of numbers, learn to write explicit and recursive formulas, and master sigma notation for compact summation — the building blocks of series and calculus.
Sequences are the foundation of series, limits, and calculus. They model everything from loan amortization schedules and population growth to the pixels rendered on your screen. Mastering explicit and recursive formulas — and the shorthand of sigma notation — gives you a precise language for describing any repeating or evolving pattern.
Essential Question
How can we describe and predict patterns in ordered lists of numbers using explicit formulas, recursive formulas, and sigma notation?
Lesson Overview
A sequence is an ordered list of numbers called terms. We write the terms as a₁, a₂, a₃, …, aₙ where the subscript tells us the position. Sequences appear everywhere — from compound interest schedules to the pixels in a digital image.
There are two main ways to define a sequence:
- Explicit formula: aₙ = f(n) — plug in any position n and get the term directly.
- Recursive formula: aₙ = f(aₙ₋₁) — each term is defined in terms of the previous one; you must know a starting value.
We also use sigma (Σ) notation to write sums compactly. The expression Σ(k=1 to n) f(k) means "add f(k) for every integer k from 1 to n."
A sequence is finite if it has a last term, and infinite if it continues without end (indicated by "…").
Worked Examples
Write the first 5 terms of the sequence defined by aₙ = 3n − 1.
Substitute n = 1: a₁ = 3(1) − 1 = 2
Substitute n = 2: a₂ = 3(2) − 1 = 5
Substitute n = 3: a₃ = 3(3) − 1 = 8
Substitute n = 4: a₄ = 3(4) − 1 = 11
Substitute n = 5: a₅ = 3(5) − 1 = 14
Write the first 5 terms of the recursive sequence: a₁ = 2, aₙ = aₙ₋₁ + 4.
Start with a₁ = 2 (given).
a₂ = a₁ + 4 = 2 + 4 = 6
a₃ = a₂ + 4 = 6 + 4 = 10
a₄ = a₃ + 4 = 10 + 4 = 14
a₅ = a₄ + 4 = 14 + 4 = 18
Find an explicit formula for the sequence 5, 8, 11, 14, 17.
Find the common difference: 8 − 5 = 3, so d = 3.
The sequence increases by 3 each step, so the formula has the form aₙ = 3n + c.
Use n = 1: a₁ = 3(1) + c = 5 → c = 2.
Check n = 2: 3(2) + 2 = 8 ✓; n = 3: 3(3) + 2 = 11 ✓
Evaluate Σ(k=1 to 4) of (2k + 1).
Write out each term by substituting k = 1, 2, 3, 4.
k = 1: 2(1) + 1 = 3
k = 2: 2(2) + 1 = 5
k = 3: 2(3) + 1 = 7
k = 4: 2(4) + 1 = 9
Sum: 3 + 5 + 7 + 9 = 24
Consider the sequence 1, 1/2, 1/4, 1/8, … Is it finite or infinite? Write sigma notation for the sum of the first 4 terms.
The '…' indicates the sequence continues without end — it is infinite.
Each term is (1/2)^(n−1): a₁=1, a₂=1/2, a₃=1/4, a₄=1/8.
The general term is aₙ = (1/2)^(n−1).
Sigma notation for the sum of the first 4 terms: Σ(k=1 to 4) (1/2)^(k−1).
Guided Practice
Write the first 5 terms of the sequence defined by aₙ = n² − 2.
Hint: Substitute n = 1, 2, 3, 4, 5 into the formula one at a time.
Write the first 4 terms of the recursive sequence: a₁ = 3, aₙ = 2aₙ₋₁ − 1.
Hint: Start with a₁ = 3, then apply the rule aₙ = 2aₙ₋₁ − 1 repeatedly.
Find an explicit formula for the sequence 4, 7, 10, 13, 16.
Hint: Find the common difference first, then use a₁ to solve for the constant.
Evaluate Σ(k=1 to 5) k².
Hint: Write out k² for k = 1, 2, 3, 4, 5 and add the results.
Write sigma notation for the sum 3 + 6 + 9 + 12 + 15.
Hint: Each term is a multiple of 3. Express the kth term as 3k and identify the bounds.
Common Mistakes
Using aₙ = aₙ₋₁ + d without a starting value for a recursive formula.
Always state the initial term (e.g., a₁ = 2) when writing a recursive formula.
Confusing the index variable in sigma notation with the sequence subscript.
The index k in Σ is a dummy variable — it only lives inside the sum.
Writing aₙ = 3n when the first term is not 3 (e.g., sequence starts at 5).
Check your formula against a₁ and adjust the constant: aₙ = 3n + c.
Assuming every sequence with '…' is infinite.
Context matters — '…' can mean 'continuing to a stated last term' (finite) or 'forever' (infinite).
Math Tips
To find an explicit formula for an arithmetic sequence, use aₙ = a₁ + (n − 1)d.
Sigma notation Σ(k=m to n) f(k) means sum f(k) for every integer k from m to n inclusive.
A recursive formula always needs two pieces: the rule AND the initial condition.
When evaluating a sigma sum, list every term — it prevents arithmetic errors.
Sequence
An ordered list of numbers, each called a term.
Explicit formula
A formula aₙ = f(n) that gives any term directly from its position n.
Recursive formula
A formula that defines each term using one or more previous terms, plus an initial condition.
Sigma notation
Compact notation Σ(k=m to n) f(k) representing the sum of f(k) over consecutive integers.
Finite sequence
A sequence with a definite last term.
Infinite sequence
A sequence that continues without end, indicated by '…'.
Interactive Practice — 5 Questions
Which of the following is an explicit formula?
What is the 4th term of aₙ = 2n + 3?
For the recursive sequence a₁ = 1, aₙ = aₙ₋₁ + 5, what is a₃?
Evaluate Σ(k=1 to 3) (k + 2).
Which sequence is infinite?