What Is an Arithmetic Sequence?
An arithmetic sequence is an ordered list of numbers where each term is obtained by adding a fixed constant to the previous one. That fixed constant is the common difference, denoted $d$.
Example sequence: $3, 7, 11, 15, 19, \ldots$ — common difference $d = 4$.
The general form is:
$$a_1,; a_1 + d,; a_1 + 2d,; a_1 + 3d,; \ldots$$
where $a_1$ is the first term and $d$ is the common difference. A sequence is arithmetic if and only if the difference between every pair of consecutive terms is the same.
What Is the Arithmetic Sequence Formula?
There are two formulas you need.
The nth-Term Formula
The $n$-th term is given by:
$$a_n = a_1 + (n - 1) d$$
where $a_1$ is the first term, $d$ is the common difference, and $n$ is the position of the term you want.
Example. Find the 20th term of $5, 8, 11, 14, \ldots$.
Here $a_1 = 5$, $d = 3$, $n = 20$:
$$a_{20} = 5 + (20 - 1)(3) = 5 + 57 = 62$$
The Sum Formula (Arithmetic Series)
The sum of the first $n$ terms — called an arithmetic series and written $S_n$ — has two equivalent forms:
$$S_n = \frac{n}{2}\bigl[2 a_1 + (n - 1)d\bigr] \quad \text{or equivalently} \quad S_n = \frac{n}{2}\bigl(a_1 + a_n\bigr)$$
The second form reads as "average of the first and last term, times the number of terms." It's faster when you already know the last term.
Example. Sum the first 20 terms of $5, 8, 11, 14, \ldots$. We just found $a_{20} = 62$:
$$S_{20} = \frac{20}{2}(5 + 62) = 10 \times 67 = 670$$
How Do You Find the Common Difference?
Subtract any term from the next one. For the sequence $a_1, a_2, a_3, \ldots$:
$$d = a_2 - a_1 = a_3 - a_2 = \cdots$$
If the difference is the same every time, the sequence is arithmetic. If the difference varies, it's not arithmetic — it might be geometric (constant ratio instead of constant difference) or something more complex.
Why Does the Sum Formula Work? (The Gauss Story)
"If we add the numbers 1 through 100 in pairs from the outside in, every pair sums to 101…" — Carl Friedrich Gauss, age 7, supposedly.
The most famous origin story in mathematics belongs to this exact formula. As a small child — usually told as age 7 or 8 — Carl Friedrich Gauss was set the busywork task of summing $1 + 2 + 3 + \cdots + 100$ by a teacher who wanted to keep the class quiet. Gauss returned with the answer 5050 in seconds.
His trick: write the sum forwards and backwards, then add column-by-column:
$$\begin{aligned}S &= 1 + 2 + 3 + \cdots + 99 + 100 \\S &= 100 + 99 + 98 + \cdots + 2 + 1 \\2S &=\underbrace{101 + 101 + \cdots + 101}_{100 \text{ times}} \\S &= \frac{100 \times 101}{2} = 5050\end{aligned}$$
This is exactly the modern formula $S_n = \frac{n}{2}(a_1 + a_n)$ — "average of first and last, times the count." Gauss had rediscovered, at age 7, a formula that had been known for centuries but never with such clean reasoning.
The result generalises to any arithmetic sequence: pair the first and last term (they sum to $a_1 + a_n$), then the second and second-to-last (same sum), and so on — $n/2$ pairs total.
Where Do Arithmetic Sequences Show Up in Real Life?
Stadium seating. Many stadiums add a fixed number of extra seats per row going up — the row capacities form an arithmetic sequence.
Salary increments. A fixed annual raise — $5,000 each year — turns annual salaries into an arithmetic sequence.
Simple interest. Earning a fixed dollar amount per year (not compound) gives an arithmetic sequence of balances.
Stacking blocks. Counting bricks in each row of a triangular stack.
Theatre seating. Rows often have a fixed-seat difference per row.
Library shelving and packaging. Counting items in widening rows.
Walking paces and stair steps. Each rise is the same height — the cumulative height is an arithmetic series.
Compound interest, on the other hand, is geometric — each year multiplies by $(1 + r)$, not adds a constant. The distinction is the entire reason banks prefer compounding for loans and consumers prefer it for savings.
What Is the Difference Between Arithmetic and Geometric Progressions?
An arithmetic progression (AP) adds a constant; a geometric progression (GP) multiplies by a constant. Both are the most common sequence types in school math, and identifying which one you have is the first step in every problem.
Feature | Arithmetic Progression (AP) | Geometric Progression (GP) |
|---|---|---|
Rule | Add a constant each step | Multiply by a constant each step |
Constant called | Common difference, $d$ | Common ratio, $r$ |
nth term | $a_n = a_1 + (n-1)d$ | $a_n = a_1 \cdot r^{n-1}$ |
Sum of first $n$ | $S_n = \frac{n}{2}(a_1 + a_n)$ | $S_n = a_1 \cdot \frac{r^n - 1}{r - 1}$, $r \neq 1$ |
Growth shape | Linear | Exponential |
Example | $3, 7, 11, 15, 19, \ldots$ ($d = 4$) | $3, 6, 12, 24, 48, \ldots$ ($r = 2$) |
Real-world | Simple interest, salary raises by fixed amount | Compound interest, population growth, viral spread |
Quick test. Given a sequence:
If the differences between consecutive terms are equal → AP.
If the ratios between consecutive terms are equal → GP.
If neither, the sequence is something else (harmonic, Fibonacci, quadratic, etc.).
Worked example. Classify $4, 12, 36, 108, \ldots$.
Differences: $12-4 = 8$, $36-12 = 24$, $108-36 = 72$ — not constant. Ratios: $12/4 = 3$, $36/12 = 3$, $108/36 = 3$ — constant.
The sequence is geometric with $r = 3$.
The cleanest mental picture: an AP traces a straight line if you plot term values vs position; a GP traces an exponential curve. Banks use this difference deliberately — simple-interest loans are AP, compound-interest savings are GP, and the GP wins over long horizons.
A Worked Example
Find the 15th term of the arithmetic sequence $4, 9, 14, 19, \ldots$.
The intuitive (wrong) approach. A student in a hurry writes the nth-term formula as $a_n = a_1 + n \cdot d$ — missing the $(n - 1)$ correction.
With $a_1 = 4$, $d = 5$, $n = 15$:
$$a_{15} \stackrel{?}{=} 4 + 15 \cdot 5 = 4 + 75 = 79$$
Listing the terms shows the 15th term is actually $74$, not $79$.
Why it fails. The first term is already at position 1 — so getting to position $n$ adds the common difference $(n - 1)$ times, not $n$ times. Counting one extra step is the universal arithmetic-sequence trap.
The correct method.
$$a_n = a_1 + (n - 1), d$$
$$a_{15} = 4 + (15 - 1)(5) = 4 + 70 = \boxed{74}$$
Check: $4, 9, 14, 19, 24, 29, 34, 39, 44, 49, 54, 59, 64, 69, 74$ — the 15th term is 74 ✓.
At Bhanzu, our trainers teach 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 Arithmetic 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: $a_{10} = 4 + 10 \cdot 5 = 54$ for the sequence starting at 4 with $d = 5$.
The correct way: $a_{10} = 4 + (10-1) \cdot 5 = 49$. The first term is already counted; you add $d$ only $n-1$ times to reach the $n$th term.
Mistake 2: Confusing common difference with common ratio
Where it slips in: Identifying the wrong sequence type. A common ratio means geometric, not arithmetic.
Don't do this: Calling $3, 6, 12, 24, \ldots$ arithmetic. (It's geometric — multiply by 2 each time, not add.)
The correct way: Check both. Differences: $6-3=3$, $12-6=6$, $24-12=12$ — not constant. Ratios: $6/3=2$, $12/6=2$, $24/12=2$ — constant. The sequence is geometric. The memorizer who only checks one type often picks wrong.
Mistake 3: Using the wrong sum formula form when the last term isn't given
Where it slips in: Asked to find $S_{100}$ for $3, 7, 11, \ldots$. Student tries $S_n = \frac{n}{2}(a_1 + a_n)$ without first computing $a_{100}$.
Don't do this: Plugging into $\frac{100}{2}(3 + a_{100})$ without finding $a_{100}$ first.
The correct way: Either find $a_{100} = 3 + 99 \cdot 4 = 399$, then $S_{100} = 50 \cdot 402 = 20{,}100$ — or use the all-in-one form $S_n = \frac{n}{2}[2a_1 + (n-1)d]$ directly. The second-guesser who pauses to ask "do I know the last term?" is asking the right question.
The Mathematicians Who Shaped Arithmetic Sequences
Carl Friedrich Gauss (1777–1855, Germany) — As a child of seven or eight, rediscovered the sum formula by pairing terms in $1+2+\cdots+100 = 5050$. Gauss became one of the greatest mathematicians in history, with contributions to number theory, statistics, astronomy, and electromagnetism.
Aryabhata (476–550 CE, India) — His treatise Aryabhatiya (499 CE) gave general formulas for the sum of an arithmetic progression, predating Gauss's rediscovery by more than a millennium.
Pythagoras (c. 570–c. 495 BCE, Greece) — The Pythagoreans studied arithmetic progressions as one of three classical proportions (arithmetic, geometric, harmonic), connecting them to musical scales and number theory.
A Practical Next Step
Try these three before going to geometric sequences.
Find the 25th term of $7, 11, 15, 19, \ldots$.
Sum the first 30 terms of $2, 5, 8, 11, \ldots$.
Given $a_3 = 12$ and $a_7 = 28$, find $a_1$ and $d$.
If problem 1 felt off by one, return to the wrong-path-first example — the $(n-1)$ is the trap. Want a live Bhanzu trainer to walk through more sequence problems? Book a free demo class — online globally.
Was this article helpful?
Your feedback helps us write better content