Arithmetic Progressions — nth Term, Sum, Examples

What is an Arithmetic Progression?

An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. That constant is the common difference, written $d$. Each new term is the one before it plus $d$.

So $3, 7, 11, 15, \dots$ is an AP with first term $a = 3$ and common difference $d = 4$. You can check $d$ by subtracting any term from the one after it: $7 - 3 = 4$, $11 - 7 = 4$. If that subtraction gives the same answer everywhere, you have an AP.

A general AP looks like this:

$a, ; a + d, ; a + 2d, ; a + 3d, ; \dots$

where $a$ is the first term. Note that $d$ can be negative (a decreasing AP like $20, 17, 14, \dots$) or zero (a constant sequence). It just has to be the same every step.

This is one branch of the wider family of sequences in algebra; its multiplicative cousin, where you multiply by a fixed ratio instead of adding, is the geometric progression.

The Arithmetic Progression Formulas

Two formulas do almost all the work. Here is what each variable means before we use them: $a$ is the first term, $d$ is the common difference, $n$ is the term number you want, $a_n$ is the value of that term, $S_n$ is the sum of the first $n$ terms, and $l$ is the last term.

nth term (general term):

$$a_n = a + (n-1)d$$

Sum of the first $n$ terms (when you know $a$ and $d$):

$$S_n = \frac{n}{2}\big[2a + (n-1)d\big]$$

Sum of the first $n$ terms (when you know the first and last terms):

$$S_n = \frac{n}{2}\big(a + l\big)$$

The two sum formulas are the same rule wearing different clothes — substitute $l = a + (n-1)d$ into the second and you recover the first.

Where the nth-term formula comes from

It is just repeated addition counted carefully. To reach the 1st term you add $d$ zero times; the 2nd term, once; the $n$th term, $(n-1)$ times:

$a_1 = a$

$a_2 = a + d$

$a_3 = a + 2d$

$a_n = a + (n-1)d$

The exponent of the pattern is $(n-1)$, not $n$, because the first term already sits there before any $d$ is added.

Where the sum formula comes from (Gauss's trick)

Write the sum forwards, then write it again backwards underneath:

$S_n = a + (a+d) + \dots + l$

$S_n = l + (l-d) + \dots + a$

Add the two lines term by term. Every pairing gives the same total, $a + l$, and there are $n$ of them:

$2S_n = n(a + l)$

$S_n = \dfrac{n}{2}(a + l)$

That is Gauss's classroom shortcut, written as algebra.

Examples of Arithmetic Progressions

Six problems, easier to harder, using the formulas above.

Example 1

Find the common difference of $5, 8, 11, 14, \dots$

Subtract consecutive terms: $8 - 5 = 3$ and $11 - 8 = 3$.

Final answer: $d = 3$.

Example 2

Find the 10th term of $7, 12, 17, 22, \dots$

A natural first move is to write $a_{10} = a + nd = 7 + 10(5) = 57$.

Why that is wrong. Using $n$ instead of $(n-1)$ adds one extra step of $d$. To reach the 10th term from the first, you add $d$ only nine times, not ten.

The correct method. Here $a = 7$, $d = 5$, $n = 10$:

$a_{10} = a + (n-1)d = 7 + (10-1)(5)$

$a_{10} = 7 + 45 = 52$

Final answer: $52$. The $(n-1)$, not $n$, is the whole game.

Example 3

Which term of $3, 8, 13, 18, \dots$ equals $98$?

Here $a = 3$, $d = 5$, and $a_n = 98$. Solve for $n$:

$98 = 3 + (n-1)(5)$

$95 = (n-1)(5)$

$n - 1 = 19$

$n = 20$

Final answer: the 20th term.

Example 4

Find the sum of the first 20 terms of $4, 9, 14, 19, \dots$

Here $a = 4$, $d = 5$, $n = 20$. Use $S_n = \frac{n}{2}[2a + (n-1)d]$:

$S_{20} = \frac{20}{2}\big[2(4) + (20-1)(5)\big]$

$S_{20} = 10\big[8 + 95\big]$

$S_{20} = 10 \times 103 = 1030$

Final answer: $1030$.

Example 5

Add all the integers from 1 to 100.

This is an AP with $a = 1$, $l = 100$, $n = 100$. Use the first-and-last form:

$S_{100} = \frac{n}{2}(a + l) = \frac{100}{2}(1 + 100)$

$S_{100} = 50 \times 101 = 5050$

Final answer: $5050$ — Gauss's result, in one line.

Example 6

A starting salary is $30{,}000$, rising by $2{,}000$ each year. What is the total earned over 10 years?

The yearly salaries form an AP: $a = 30000$, $d = 2000$, $n = 10$.

$S_{10} = \frac{10}{2}\big[2(30000) + (10-1)(2000)\big]$

$S_{10} = 5\big[60000 + 18000\big]$

$S_{10} = 5 \times 78000 = 390000$

Final answer: $390{,}000$ earned over the decade.

Where Arithmetic Progressions Show Up

The constant-step pattern is everywhere money, time, or stock changes by a fixed amount.

• Salaries and savings: A fixed annual raise or a fixed monthly deposit is an AP; the sum formula tells you the lifetime total without adding year by year.

• Loan repayment and depreciation: A fixed reduction each period — paying off a set amount, or an asset losing the same value yearly — traces an AP.

• Seating, stacking, and scheduling: Theatre rows that grow by a fixed number of seats, logs stacked in a tapering pile, evenly spaced event times — all APs, all solvable with $S_n$.

The pairing trick that powers the sum formula is usually credited to Carl Friedrich Gauss (1777–1855, Germany), who reportedly used it as a schoolboy; you can read his biography at MacTutor. The deeper point is that an AP turns "add up a long list" into "multiply two numbers" — that compression is why the formula has survived for centuries.

Common Mistakes With Arithmetic Progressions

Mistake 1: Using $n$ instead of $(n-1)$ in the nth-term formula

Where it slips in: Computing a specific term. The rusher reads "10th term" and multiplies $d$ by 10.

Don't do this: $a_{10} = a + 10d$.

The correct way: $a_{10} = a + 9d$. You add the common difference one fewer time than the term number, because the first term is already in place before any $d$ is added. The fix that sticks: always write $(n-1)$ explicitly before substituting a number.

Mistake 2: Getting the common difference backwards

Where it slips in: Decreasing sequences. The second-guesser computes $a_1 - a_2$ instead of $a_2 - a_1$ and loses the sign.

Don't do this: For $20, 17, 14, \dots$ write $d = 20 - 17 = 3$.

The correct way: $d = a_2 - a_1 = 17 - 20 = -3$. The common difference is always "later term minus earlier term." A negative $d$ is correct and expected for a decreasing AP — do not drop the minus sign.

Mistake 3: Picking the wrong sum formula

Where it slips in: You are given the last term but reach for the $[2a + (n-1)d]$ version anyway, then have to back out $d$.

Don't do this: Compute $d$ from scratch when you already know $a$, $l$, and $n$.

The correct way: When the last term $l$ is known, use $S_n = \frac{n}{2}(a + l)$ directly — it is faster and avoids an extra step where errors creep in. Choosing the formula that matches your given information is half of doing these cleanly. Errors of this kind compound in applied settings; a fixed-step financial schedule miscounted by one period is the everyday analogue of a Mars Climate Orbiter-style off-by-one feeding the wrong total into the next stage.

Conclusion

• An arithmetic progression adds a constant common difference $d$ at every step, giving the form $a, a+d, a+2d, \dots$

• The nth term is $a_n = a + (n-1)d$ — note the $(n-1)$, not $n$.

• The sum is $S_n = \frac{n}{2}[2a + (n-1)d]$, or $S_n = \frac{n}{2}(a + l)$ when the last term is known.

• The single most common mistake is using $n$ instead of $(n-1)$ in the term formula.

• APs model anything that changes by a fixed amount — salaries, deposits, stacks, schedules.

Practice These to Solidify Your Understanding

Work through each, then revisit the linked section if you stall.

1. Find the 15th term of $6, 10, 14, 18, \dots$ (Revisit Example 2 and Mistake 1.)

2. Find the sum of the first 30 terms of $5, 9, 13, \dots$ (Revisit Example 4.)

3. Which term of $100, 95, 90, \dots$ equals $0$? (Revisit Example 3 and Mistake 2.)

To work through more progressions with a teacher, explore Bhanzu's algebra tutor or browse math tutoring options. Want a live Bhanzu trainer to walk through more AP problems? Book a free demo class.

Read More

• Common difference — a deeper look at the value of $d$ and how to find it.

• Sum of arithmetic sequence formula — the sum formula on its own, with more derivations.

• Geometric sequence — the multiply-by-a-ratio counterpart.

• Sigma notation — compact notation for writing the sum of an AP.

• Progression in maths — how arithmetic, geometric, and harmonic progressions relate.