Sequences in Algebra - Types, Formulas, Examples
Sequences in Algebra - Types, Formulas, Examples
What Is a Sequence in Algebra?
A sequence in algebra is an ordered list of numbers that follows a specific rule. Each number in the list is called a term. The first term is a1, the second is a2, and the n-th is an.
A few examples:
- 2, 4, 6, 8, 10,… — add 2 each time (arithmetic)
- 3, 6, 12, 24, 48,… — multiply by 2 each time (geometric)
- 1, 1, 2, 3, 5, 8, 13,… — add the two previous (Fibonacci)
- 1, 4, 9, 16, 25,… — square numbers
The point of studying sequences is to find the rule — the formula or recurrence that produces any term you want, without writing out the entire list. That formula is called the nth-term formula or general term.
What Is the Order of the Sequence?
The order of a sequence is the position of each term — its place in the ordered list. The first term is at position n=1, the second at n=2, and so on.
A sequence's order matters: 1, 2, 3, 4 and 4, 3, 2, 1 are different sequences even though they contain the same numbers. (Unlike a set, where order doesn't matter.)
Sequences can be in ascending order (each term larger than the previous) or descending order (each term smaller). They can also be neither — Fibonacci is ascending; an alternating sequence like 1, −1, 1, −1,… is neither.
What Are Finite and Infinite Sequences?
A sequence is finite if it has a last term — the list eventually stops. It is infinite if it continues forever (no last term).
Finite Sequences
A finite sequence has a definite number of terms. Examples:
- The days of the week numbered 1, 2, 3, 4, 5, 6, 7 — a finite sequence of 7 terms.
- The first 10 even numbers 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 — finite, 10 terms.
- The integers from 1 to 100 — finite, 100 terms.
Infinite Sequences
An infinite sequence has no last term — it continues indefinitely. Examples:
- Natural numbers 1, 2, 3, 4, 5,… — never ends.
- Squares 1, 4, 9, 16, 25,… — never ends.
- The decimal expansion of π: 3, 1, 4, 1, 5, 9, 2, 6,… — provably never ends.
What Are the 8 Main Types of Sequences?
1. Arithmetic Sequence (Arithmetic Progression / AP)
The difference between consecutive terms is constant. That constant is the common difference, d.
Example: 5, 8, 11, 14, 17,… — common difference d=3.
The nth-term formula:
an = a1 + (n−1)d
2. Geometric Sequence (Geometric Progression / GP)
The ratio between consecutive terms is constant. That constant is the common ratio, r.
Example: 3, 6, 12, 24, 48,… — common ratio r=2.
The nth-term formula:
an = a1 ⋅ r^{n−1}
3. Harmonic Sequence
The reciprocals of the terms form an arithmetic sequence.
Example: 1, 1/2, 1/3, 1/4, 1/5,… — the reciprocals 1, 2, 3, 4, 5,… form an arithmetic sequence with d=1.
The nth-term formula:
an = 1 / (a + (n−1)d)
4. Fibonacci Sequence
A recursive sequence where each term is the sum of the previous two.
F1=1, F2=1, Fn=Fn−1+Fn−2
First ten terms: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,…
5. Triangular Number Sequence
Each term counts the number of dots needed to form a triangle.
1, 3, 6, 10, 15, 21, 28, 36, 45, 55,…
The nth-term formula:
Tn = n(n+1)/2
6. Square Number Sequence
Each term is a perfect square of its position.
1, 4, 9, 16, 25, 36, 49, 64, 81, 100,…
The nth-term formula:
Sn = n^2
7. Cube Number Sequence
Each term is the cube of its position.
1, 8, 27, 64, 125, 216, 343, 512,…
The nth-term formula:
Cn = n^3
8. Quadratic Sequence
A sequence whose second differences are constant.
Example: 2, 5, 10, 17, 26,… (first differences: 3, 5, 7, 9,…; second differences: 2, 2, 2,…)
The nth-term formula is:
an = an^2 + bn + c
What Are Series and Partial Sums of Sequences?
A series is the sum of the terms of a sequence. If the sequence is a1, a2, a3,…, the series is:
a1 + a2 + a3 + ⋯
A partial sum is the sum of the first n terms:
Sn = a1 + a2 + a3 + ⋯ + an
Using sigma notation:
Sn = ∑_{k=1}^{n} ak
Sum of an Arithmetic Series
Sn = n/2[2a1 + (n−1)d]
Sum of a Geometric Series (Finite)
Sn = a1(1−r^n)/(1−r), r≠1
Sum of an Infinite Geometric Series (|r|<1)
S∞ = a1/(1−r)
What Are the Rules of Sequences? (Explicit vs Recursive)
Explicit Rule (Closed Form)
An explicit rule gives an directly as a function of n — you can jump to any term without computing the ones before it.
Examples:
- Arithmetic: an = a1 + (n−1)d
- Geometric: an = a1 r^{n−1}
Recursive Rule (Recurrence Relation)
A recursive rule defines each term in terms of one or more previous terms — plus initial conditions.
Examples:
- Arithmetic: an = an−1 + d, with a1 given.
- Geometric: an = an−1 ⋅ r, with a1 given.
- Fibonacci: Fn = Fn−1 + Fn−2, with F1 = 1, F2 = 1.
What Are the Most Common Mistakes With Sequences?
Mistake 1: Off-by-one in the nth-term formula
Where it slips in: Writing an = a1 + n ⋅ d instead of an = a1 + (n−1)d.
Mistake 2: Confusing arithmetic with geometric
Where it slips in: Spotting a pattern but classifying it wrong.
Mistake 3: Treating a quadratic sequence as arithmetic
Where it slips in: Sequences like 2, 6, 12, 20 — the second differences are constant, but the first differences aren't.
Where Do Sequences Appear in Real Life?
- Compound interest.
- Population growth and radioactive decay.
- Sunflower seeds and pinecones.
- Music — overtones.
- Loan amortisation.
- Handshakes.
Frequently Asked Questions
What is a sequence in algebra? A sequence is an ordered list of numbers following a rule.
What are the main types of sequences? The eight most studied: arithmetic, geometric, harmonic, Fibonacci, triangular, square, cube, and quadratic.
What is a series in math? A series is the sum of the terms of a sequence.