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 $a_1$, the second is $a_2$, and the $n$-th is $a_n$.

A few examples:

• $2, 4, 6, 8, 10, \ldots$ — add 2 each time (arithmetic)

• $3, 6, 12, 24, 48, \ldots$ — multiply by 2 each time (geometric)

• $1, 1, 2, 3, 5, 8, 13, \ldots$ — add the two previous (Fibonacci)

• $1, 4, 9, 16, 25, \ldots$ — 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.

$$\underbrace{a_1}{n=1}, ;; \underbrace{a_2}{n=2}, ;; \underbrace{a_3}{n=3}, ;; \ldots, ;; \underbrace{a_n}{n\text{-th}}, ;; \ldots$$

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 $1, -1, 1, -1, \ldots$ 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.

Written: $a_1, a_2, a_3, \ldots, a_n$ — the $\ldots$ stops at $a_n$.

Infinite Sequences

An infinite sequence has no last term — it continues indefinitely. Examples:

• Natural numbers $1, 2, 3, 4, 5, \ldots$ — never ends.

• Squares $1, 4, 9, 16, 25, \ldots$ — never ends.

• The decimal expansion of $\pi$: $3, 1, 4, 1, 5, 9, 2, 6, \ldots$ — provably never ends.

Written: $a_1, a_2, a_3, \ldots$ — the $\ldots$ never stops.

Many real-world sequences are modelled as infinite even when, in practice, they terminate — bacterial growth, half-lives, compound interest. The math of infinite sequences (limits, convergence) is the gateway to calculus.

What Are the 8 Main Types of Sequences?

The most-studied types of sequences in school and early college mathematics.

1. Arithmetic Sequence (Arithmetic Progression / AP)

The difference between consecutive terms is constant. That constant is the common difference, $d$.

Example: $5, 8, 11, 14, 17, \ldots$ — common difference $d = 3$.

The nth-term formula:

$$a_n = a_1 + (n-1), d$$

2. Geometric Sequence (Geometric Progression / GP)

The ratio between consecutive terms is constant. That constant is the common ratio, $r$.

Example: $3, 6, 12, 24, 48, \ldots$ — common ratio $r = 2$.

The nth-term formula:

$$a_n = a_1 \cdot r^{n-1}$$

3. Harmonic Sequence

The reciprocals of the terms form an arithmetic sequence.

Example: $1, \tfrac{1}{2}, \tfrac{1}{3}, \tfrac{1}{4}, \tfrac{1}{5}, \ldots$ — the reciprocals $1, 2, 3, 4, 5, \ldots$ form an arithmetic sequence with $d = 1$.

The nth-term formula:

$$a_n = \frac{1}{a + (n-1) d} \quad \text{where } a, d \text{ are the first term and common difference of the underlying AP}$$

Harmonic sequences appear in music (the harmonic series of overtones) and physics (vibrating strings).

4. Fibonacci Sequence

A recursive sequence where each term is the sum of the previous two.

$$F_1 = 1, \quad F_2 = 1, \quad F_n = F_{n-1} + F_{n-2}$$

First ten terms: $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \ldots$

Introduced to Europe in 1202 by Leonardo of Pisa (Fibonacci) in Liber Abaci. The closed-form (Binet's formula):

$$F_n = \frac{\varphi^n - (1 - \varphi)^n}{\sqrt{5}} \quad \text{where } \varphi = \frac{1 + \sqrt{5}}{2}$$

5. Triangular Number Sequence

Each term counts the number of dots needed to form a triangle.

$$1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \ldots$$

The nth-term formula:

$$T_n = \frac{n(n+1)}{2}$$

Discovered by the Pythagoreans in ancient Greece. Used in counting handshakes (5 people → 10 handshakes), staircase blocks, and pool-rack arrangements.

6. Square Number Sequence

Each term is a perfect square of its position.

$$1, 4, 9, 16, 25, 36, 49, 64, 81, 100, \ldots$$

The nth-term formula:

$$S_n = n^2$$

The differences between consecutive squares form the odd numbers ($3, 5, 7, 9, \ldots$) — a beautiful side-result.

7. Cube Number Sequence

Each term is the cube of its position.

$$1, 8, 27, 64, 125, 216, 343, 512, \ldots$$

The nth-term formula:

$$C_n = n^3$$

The sum of the first $n$ cubes equals the square of the sum of the first $n$ natural numbers: $\sum_{k=1}^{n} k^3 = \left(\frac{n(n+1)}{2}\right)^2$.

8. Quadratic Sequence

A sequence whose second differences are constant — the difference of the differences is the same.

Example: $2, 5, 10, 17, 26, \ldots$. First differences: $3, 5, 7, 9, \ldots$. Second differences: $2, 2, 2, \ldots$ (constant).

The nth-term formula is always quadratic in $n$:

$$a_n = an^2 + bn + c$$

For the example above, $a_n = n^2 + 1$: terms are $2, 5, 10, 17, 26$ ✓.

What Are Series and Partial Sums of Sequences?

A series is the sum of the terms of a sequence. If the sequence is $a_1, a_2, a_3, \ldots$, the series is:

$$a_1 + a_2 + a_3 + \cdots$$

A partial sum is the sum of the first $n$ terms, written $S_n$:

$$S_n = a_1 + a_2 + a_3 + \cdots + a_n$$

Using sigma notation (the capital Greek letter $\Sigma$):

$$S_n = \sum_{k=1}^{n} a_k$$

Sum of an Arithmetic Series

$$S_n = \frac{n}{2}\bigl[2 a_1 + (n - 1) d\bigr] = \frac{n}{2}\bigl(a_1 + a_n\bigr)$$

(Famous as the formula Gauss rediscovered as a child by pairing $1 + 100, 2 + 99, \ldots$.)

Sum of a Geometric Series (Finite)

$$S_n = \frac{a_1 (1 - r^n)}{1 - r}, \quad r \neq 1$$

Sum of an Infinite Geometric Series ($|r| < 1$)

$$S_\infty = \frac{a_1}{1 - r}$$

This is why the infinite sum $\tfrac{1}{2} + \tfrac{1}{4} + \tfrac{1}{8} + \cdots$ equals exactly 1 — Zeno's paradox of Achilles and the tortoise, resolved by one formula.

Sum of a Triangular Sequence

$$T_1 + T_2 + \cdots + T_n = \binom{n+2}{3} = \frac{n(n+1)(n+2)}{6}$$

What Are the Rules of Sequences? (Explicit vs Recursive)

Every sequence can be described in one of two ways.

Explicit Rule (Closed Form)

An explicit rule gives $a_n$ directly as a function of $n$ — you can jump to any term without computing the ones before it.

Examples:

• Arithmetic: $a_n = a_1 + (n-1)d$

• Geometric: $a_n = a_1 r^{n-1}$

• Square: $a_n = n^2$

Recursive Rule (Recurrence Relation)

A recursive rule defines each term in terms of one or more previous terms — plus initial conditions.

Examples:

• Arithmetic: $a_n = a_{n-1} + d$, with $a_1$ given.

• Geometric: $a_n = a_{n-1} \cdot r$, with $a_1$ given.

• Fibonacci: $F_n = F_{n-1} + F_{n-2}$, with $F_1 = 1$, $F_2 = 1$.

Recursive rules are simple to describe but slow to compute large terms. Explicit rules require more algebra but let you jump straight to the 1000th term in one step. Both have their uses — recursive for simple recurrences like Fibonacci; explicit for fast computation.

What Are the Standard Sequence Formulas?

A quick reference of the most-used formulas.

How Do You Find Missing Numbers in a Sequence?

A classic problem: given $2, ,?, , ,8, ,?, , ,18, \ldots$, find the missing terms.

Strategy.

1. Look at differences. Compute the differences between consecutive known terms.

2. Spot the pattern. Is the difference constant (arithmetic)? Is the ratio constant (geometric)? Are the differences themselves changing in a pattern (quadratic)?

3. Apply the rule. Use the pattern to fill in missing terms.

Example 1. Find the missing term: $5, 8, ?, 14, 17$.

Differences: $8 - 5 = 3$, then $?$, then $14 - ? = 3$ (if arithmetic), then $17 - 14 = 3$. So the differences are all 3, meaning the missing term is $8 + 3 = 11$.

Sequence: $5, 8, 11, 14, 17$ ✓.

Example 2. Find the missing term: $1, 4, ?, 16, 25$.

These are square numbers: $1^2, 2^2, ?, 4^2, 5^2$. Missing term: $3^2 = 9$.

Example 3. Find the missing term: $2, 6, 12, ?, 30, 42$.

First differences: $4, 6, ?, ?, 12$. They're $4, 6, 8, 10, 12$ — increasing by 2 each time. So $?$ in first differences is $8$, and the missing term is $12 + 8 = 20$.

Sequence: $2, 6, 12, 20, 30, 42$. (These are the products $n(n+1)$ for $n = 1, 2, 3, 4, 5, 6$.)

Where Did Sequences Come From? (The Rabbit Problem)

"A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year?" — Fibonacci, Liber Abaci, 1202.

In 1202 CE, the Italian mathematician Leonardo of Pisa — better known as Fibonacci — published Liber Abaci ("The Book of Calculation"). One problem asked about rabbit reproduction: starting with a pair of newborn rabbits, each pair produces a new pair every month from the second month onward — how many pairs are there after $n$ months?

The answer is the famous sequence $1, 1, 2, 3, 5, 8, 13, \ldots$ — each term the sum of the previous two. Fibonacci posed it as a puzzle; the deeper structure of why this sequence shows up everywhere in nature wasn't understood until centuries later.

Triangular numbers were studied by the Pythagoreans (~500 BCE), who treated them as one of three classical figurate number sequences (triangular, square, oblong). The arithmetic-series sum formula was rediscovered by Carl Friedrich Gauss as a child of seven — pairing terms in $1 + 2 + \cdots + 100 = 5050$ in seconds.

By the 18th century, Leonhard Euler had formalised the connection between sequences, series, and limits — laying the foundation for modern calculus.

Where Do Sequences Appear in Real Life?

• Compound interest. $1{,}000$ invested at 5% annually becomes $1{,}000 \cdot 1.05^n$ after $n$ years — a geometric sequence.

• Population growth and radioactive decay. Both geometric. Carbon-14 dating uses the half-life sequence $N(t) = N_0 \cdot (1/2)^{t/5730}$.

• Sunflower seeds and pinecones. Fibonacci numbers govern the spiral arrangements in sunflower heads and pinecone scales.

• Music — overtones. A vibrating string produces overtones at frequencies $1, \tfrac{1}{2}, \tfrac{1}{3}, \ldots$ of the fundamental — a harmonic sequence.

• Loan amortisation. Monthly mortgage payments are computed using geometric-series sums.

• Handshakes. $n$ people at a party shake hands $T_{n-1} = \tfrac{n(n-1)}{2}$ times — triangular numbers.

• Stairs and stacked blocks. Counting bricks in widening rows — triangular or square sequences.

• Computer science — algorithm complexity. Big-O notation uses sequence growth ($n$, $n^2$, $2^n$) to describe runtimes.

A Worked Example — Wrong Path First

Find the 15th term of the arithmetic sequence $4, 9, 14, 19, 24, \ldots$.

The intuitive (wrong) approach. A student writes the formula as $a_n = a_1 + n \cdot d$ instead of $a_n = a_1 + (n-1) \cdot d$ — missing the off-by-one correction.

With $a_1 = 4$, $d = 5$, $n = 15$:

$$a_{15} \stackrel{?}{=} 4 + 15 \cdot 5 = 79$$

Listing the terms shows the 15th is actually $74$, not $79$.

Why it fails. The first term is at position 1 already — so to reach position $n$, you add $d$ a total of $(n-1)$ times, not $n$ times. Counting one extra step is the universal arithmetic-sequence trap.

The correct method.

$$a_n = a_1 + (n-1), d, \quad a_{15} = 4 + 14 \cdot 5 = \boxed{74}$$

Check. $4, 9, 14, 19, 24, 29, 34, 39, 44, 49, 54, 59, 64, 69, 74$ ✓.

At Bhanzu, our trainers walk through this wrong-path-first sequence intentionally — the rusher who skips the $(n-1)$ is the most common archetype to hit this, and the only way the rule sticks is to feel the off-by-one once.

What Are the Most Common Mistakes With Sequences?

Mistake 1: Off-by-one in the nth-term formula

Where it slips in: Writing $a_n = a_1 + nd$ instead of $a_n = a_1 + (n-1)d$.

Don't do this: For $a_1 = 4$, $d = 5$, $n = 10$: $a_{10} = 4 + 50 = 54$.

The correct way: $a_{10} = 4 + 9 \cdot 5 = 49$. Add $d$ exactly $n-1$ times to reach position $n$.

Mistake 2: Confusing arithmetic with geometric

Where it slips in: Spotting a pattern but classifying it wrong.

Don't do this: Calling $3, 6, 12, 24$ arithmetic. The differences $3, 6, 12, \ldots$ aren't constant; the ratios $2, 2, 2$ are. It's geometric.

The correct way: Always check both: constant difference → arithmetic; constant ratio → geometric. The memorizer who picks one without checking the other often picks wrong.

Mistake 3: Treating a quadratic sequence as arithmetic

Where it slips in: Sequences like $2, 6, 12, 20, 30$ — the second differences are constant, but the first differences aren't.

Don't do this: Look at $2, 6, 12, 20, 30$, see differences $4, 6, 8, 10$, and call it arithmetic.

The correct way: First differences aren't constant — but the second differences are: $2, 2, 2$. That's a quadratic sequence with formula $a_n = n(n+1) = n^2 + n$. The second-guesser who computes both first and second differences catches this.

The Mathematicians Who Shaped Sequences

Leonardo of Pisa (Fibonacci) (c. 1170–c. 1250, Italy) — His 1202 book Liber Abaci introduced the Fibonacci sequence via the rabbit problem. The sequence shows up in sunflower spirals, pinecones, and the genealogy of honey bees.

Carl Friedrich Gauss (1777–1855, Germany) — As a child of seven or eight, rediscovered the arithmetic-sum formula by pairing $1+100, 2+99, \ldots$ to compute $1+2+\cdots+100=5050$ in seconds. Became one of the greatest mathematicians in history.

Jacques Binet (1786–1856, France) — Discovered the closed-form Binet's formula for Fibonacci numbers using the golden ratio: $F_n = \frac{\varphi^n - (1-\varphi)^n}{\sqrt{5}}$.

A Practical Next Step

Try these five before moving on to limits and convergence.

1. Find the 25th term of the arithmetic sequence $7, 11, 15, 19, \ldots$.

2. Find the 6th term of the geometric sequence $2, 6, 18, 54, \ldots$.

3. Find the missing term: $3, 7, ?, 19, 27, 37$. (Hint: check second differences.)

4. Sum the first 100 natural numbers $1 + 2 + 3 + \cdots + 100$.

5. Identify the type: $1, 1, 2, 3, 5, 8, 13$. Compute $F_{10} / F_9$. How close is it to the golden ratio $\varphi \approx 1.618$?

If problem 3 felt tricky: first differences $4, ?, ?, 8, 10$. Second differences are $2, 2, 2, 2$, so the first differences are $4, 6, 8, 8, 10$ — wait, second differences should be constant. So first differences are $4, 6, 8, 10$ — and the missing first difference is $6$. Missing term: $7 + 6 = 13$. Sequence: $3, 7, 13, 19, 27, 37$. ✓ Want a live Bhanzu trainer to walk through more sequence problems? Book a free demo class — online globally.