What number comes next in 2, 5, 8? The answer is 11, because each step adds 3. That is an arithmetic sequence. Now try 3, 6, 12, where each step doubles the last number. That is a geometric sequence, and it grows much faster. This guide shows how to spot each type, find any term, and add up the terms.
| Arithmetic rule | Add the same number (d) every step |
|---|---|
| Geometric rule | Multiply by the same number (r) every step |
| Arithmetic nth term | an = a1 + (n – 1)d |
| Geometric nth term | an = a1 x r(n – 1) |
| Sum of n terms | Arithmetic: n(a1 + an) / 2. Geometric: a1(1 – rn) / (1 – r) |
| Infinite geometric sum | a1 / (1 – r), only when r is between -1 and 1 |
Sequence Formulas on One Card
The table below puts both sequence types side by side. Use it as a quick lookup, then read the sections that follow for the reasons behind each formula.
| Feature | Arithmetic | Geometric |
|---|---|---|
| Each step | Adds d | Multiplies by r |
| Test | Differences are equal | Ratios are equal |
| Example | 2, 5, 8, 11 (d = 3) | 3, 6, 12, 24 (r = 2) |
| nth term | a1 + (n – 1)d | a1 x r(n – 1) |
| 10th or 8th term | a10 = 29 | a8 = 384 |
| Sum of those terms | S10 = 155 | S8 = 765 |
| Graph shape | Straight line | Curve that bends up or levels off |
Notice the key difference in the bottom row. An arithmetic sequence plots as evenly spaced dots on a straight line. A geometric sequence plots as a curve, because each jump is bigger or smaller than the last.
How Do You Tell the Two Types Apart?
Subtract each term from the next one, then divide each term by the one before it. Equal differences mean arithmetic. Equal ratios mean geometric.
Try it on 2, 5, 8, 11. The differences are 3, 3, and 3, so the sequence is arithmetic with d = 3. For 3, 6, 12, 24, the differences are 3, 6, and 12, which do not match. The ratios are 2, 2, and 2, so that sequence is geometric with r = 2.
Some lists pass neither test. The square numbers 1, 4, 9, 16 have differences of 3, 5, and 7, and ratios that keep shrinking. Those patterns follow other rules, so the formulas in this guide do not apply to them.
- Term
- One number in the list. The first term is written a1, and the term in position n is written an.
- Common difference (d)
- The fixed amount added at each step of an arithmetic sequence. It can be negative, as in 20, 17, 14.
- Common ratio (r)
- The fixed number each term is multiplied by in a geometric sequence. A negative ratio, such as -2, makes the signs flip back and forth.
- Series
- The sum of the terms of a sequence. A partial sum Sn adds only the first n terms.
- Converge
- An infinite sum converges when its running total settles toward one fixed number instead of growing without limit.
How Do You Find Any Term Without Listing Them All?
Plug the position number into the nth-term formula. You start from the first term and apply the step n – 1 times in a single calculation.
The Arithmetic Case
For 2, 5, 8, the first term is 2 and d is 3. The 10th term is 2 + (10 – 1) x 3, which equals 2 + 27, or 29. Why n – 1 and not n? The first term already sits in place, so only 9 steps separate it from the 10th term.
The formula also works backward. Which term of this sequence equals 200? Solve 2 + (n – 1) x 3 = 200, and you get n – 1 = 66. So 200 is the 67th term.
The Geometric Case
For 3, 6, 12, the first term is 3 and r is 2. The 8th term is 3 x 27, which is 3 x 128, or 384. Listing all eight terms by hand gives the same answer: 3, 6, 12, 24, 48, 96, 192, 384.
To check your work on longer lists, the sequence calculator for nth terms and sums takes the first term, the step, and the position, then returns the term, the list, and the total.
How Do You Add Up the Terms of a Sequence?
Use n(a1 + an) / 2 for an arithmetic sum and a1(1 – rn) / (1 – r) for a geometric sum. Both formulas skip the long addition.
Pairing Up an Arithmetic Sum
The arithmetic formula comes from a pairing trick. Write 2, 5, 8, up to 29, and pair the first term with the last. Every pair adds to 31, and ten terms make five pairs. Five times 31 gives S10 = 155.
The same idea sizes a real project. A theater has 20 seats in row 1 and adds 2 seats per row for 15 rows. Row 15 holds 20 + 14 x 2 = 48 seats. The whole room holds 15 x (20 + 48) / 2 = 510 seats.
Summing a Geometric List
For 3, 6, 12, up to 384, the formula gives 3 x (1 – 28) / (1 – 2). That is 3 x (-255) / (-1), or 765. With r = 2, there is a shortcut: the sum equals the next term minus the first term, so 768 – 3 = 765.
Can an Infinite Sequence Have a Finite Sum?
Yes, a geometric sequence can, when its ratio sits between -1 and 1. The terms shrink so fast that the total settles at a1 / (1 – r).
Take 100, 50, 25, and so on, where r = 1/2. The 6th term is 100 x (1/2)5 = 3.125. The infinite sum is 100 / (1 – 0.5) = 200. After 10 terms, the running total is already 199.80.
This does not work for an arithmetic sequence with a nonzero step. Adding 3 forever, or even 0.001 forever, grows past any limit you pick. A geometric sequence with r = 2 or r = -2 also has no finite sum, because its terms keep getting larger.
The shrinking case shows up in science and daily life. A drug dose that halves every few hours and a bouncing ball that loses part of its height each bounce both follow this pattern.
Why Does Geometric Growth Pull Ahead?
Geometric growth adds a larger amount each step, while arithmetic growth adds the same amount forever. Starting from a positive number, any ratio above 1 overtakes any fixed step, given enough steps.
Compare two lists that both start at 100. One adds 10 per step, and the other grows 10 percent per step. They match at the 2nd term, since both reach 110. By the 10th term, the arithmetic list reaches 190, while the geometric list reaches about 235.8.
Money is the classic case. A balance of 1,000 growing 5 percent a year is a geometric sequence with r = 1.05. After 10 years it reaches about 1,628.89. The pattern behind that growth is explained in the mathematics of doubling time, and the effect of monthly or daily growth steps is covered in how compounding frequency affects returns.
Where Do You Meet These Patterns in Real Life?
Arithmetic sequences show up wherever a fixed amount repeats. Geometric sequences show up wherever a fixed percentage repeats.
| Situation | Type | Step |
|---|---|---|
| Saving the same amount each month | Arithmetic | Add a fixed deposit |
| Seats that grow by 2 per row | Arithmetic | d = 2 |
| Stair heights on a staircase | Arithmetic | Add one riser height |
| A balance earning 5 percent a year | Geometric | r = 1.05 |
| A quantity cut in half each period | Geometric | r = 0.5 |
| A cell count that doubles each cycle | Geometric | r = 2 |
Doubling gets huge quickly. Ten doublings from 1 reach 1,024, and twenty doublings reach 1,048,576. That speed is why a ratio of 2 feels slow at first and then explodes.
Label the step before you pick a formula. Ask one simple question: does each step add a fixed amount, or multiply by a fixed amount? That single answer tells you which formula to use.
The Sequence Calculator finds the nth term, lists the terms, and adds them up for arithmetic and geometric sequences.
Questions People Ask About Sequences
What Is the Difference Between Arithmetic and Geometric Sequences?
An arithmetic sequence adds the same number at each step, such as 2, 5, 8. A geometric sequence multiplies by the same number at each step, such as 3, 6, 12. Equal differences mean arithmetic, and equal ratios mean geometric.
How Do You Find the 10th Term of 2, 5, 8?
Use the formula a1 + (n – 1)d with a1 = 2, d = 3, and n = 10. That gives 2 + 9 x 3, which equals 29. The sum of those first 10 terms is 155.
Can a Sequence Be Both Arithmetic and Geometric?
Yes, but only a constant list with a nonzero value, such as 5, 5, 5. It adds 0 at each step and multiplies by 1 at each step. Every other sequence is one type, the other, or neither.
When Does a Geometric Series Have a Finite Sum?
An infinite geometric series has a finite sum when the ratio r sits between -1 and 1. The sum is the first term divided by 1 minus r. For 100, 50, 25, and so on, the sum is 200.
Is Compound Interest a Geometric Sequence?
Yes. A balance that grows by a fixed percentage each period is multiplied by the same ratio every time. At 5 percent a year, the ratio is 1.05, so 1,000 grows to about 1,628.89 after 10 years.
Where These Numbers Come From
References Used in This Article
This article is general math education. Every worked value was computed from the formulas shown. Reviewed for accuracy by Prof. Dr. Khalil Mudassar, PhD. Last updated September 27, 2026.
Author
Shakeel Muzaffar is the Founder and Editor-in-Chief of MultiCalculators.com, bringing over 15 years of experience in digital publishing, product strategy, and online tool development. He leads the platform's editorial vision, ensuring every calculator meets strict standards for accuracy, usability, and real-world value. Shakeel personally oversees content quality, formula verification workflows, and the platform's commitment to publishing tools that are genuinely useful for students, professionals, and everyday users worldwide.
