No calculations yet.
Your sequence history will appear here.
Arithmetic Formulas
Geometric Formulas
Whether you are completing middle school algebra homework or prepping for high-level college math exams, finding the “nth term” of a number sequence is a foundational mathematical skill. While calculating the 3rd or 4th term of a sequence by hand is easy enough, figuring out the 100th term or the 500th term manually is incredibly tedious and prone to frustrating algebraic errors.
Our free Nth Term Calculator does the heavy lifting for you. It instantly solves both Arithmetic sequences (where numbers are added or subtracted) and Geometric sequences (where numbers are multiplied or divided).
Simply enter your starting number and the sequence pattern, and the tool will instantly output the exact value of the targeted term, calculate the sum of all the terms up to that point, and generate the exact mathematical formula for your homework notes!
How to Use the Nth Term Calculator
Finding the value of any term in a sequence requires knowing three specific variables. Here is how to map your math problem to our calculator:
- Select Sequence Type: Choose whether your number pattern is Arithmetic (+ or -) or Geometric (× or ÷).
- Enter the First Term (a1): What number does your sequence start with? If your sequence is 3, 6, 9, 12… your first term is 3.
- Enter the Common Difference or Ratio: If it is an arithmetic sequence, enter the amount being added or subtracted each time (the difference). If it is a geometric sequence, enter the number you are multiplying by (the ratio).
- Enter N (The Term Number): Which specific position in the sequence are you trying to find? If you want to find the 50th number in the pattern, enter 50.
- Calculate: The tool will output the exact value of the Nth term, the sum of all previous terms, and visually map out the start of your sequence!
The Exact Math Formulas
If your teacher requires you to “show your work” on a test, you must memorize these formulas. Fortunately, our calculator generates the exact filled-in formula for your specific problem in the “Exact Sequence Formula” box!
Formulas for Arithmetic Sequences:
- Find the Nth Term: an = a1 + (n – 1)d
- Sum of First N Terms: Sn = (n / 2) * [2a1 + (n – 1)d]
Formulas for Geometric Sequences:
- Find the Nth Term: an = a1 * r^(n – 1)
- Sum of First N Terms: Sn = a1 * (1 – r^n) / (1 – r)
Step-by-Step Example
Let’s look at a real-world math problem: “Find the 20th term of the sequence: 4, 11, 18, 25…”
- Identify the Type: The numbers are going up by exactly 7 each time (11 – 4 = 7). This means it is an Arithmetic Sequence.
- Identify the First Term (a1): The sequence starts with 4.
- Identify the Common Difference (d): We are adding 7 every time.
- Identify the Target Term (n): The problem asks for the 20th term.
Now plug those numbers into the standard formula:
- an = a1 + (n – 1)d
- a20 = 4 + (20 – 1)7
- a20 = 4 + (19 * 7)
- a20 = 4 + 133
- The 20th term is 137.
(If you type those exact numbers into our calculator above, you will get 137 instantly, plus it will tell you that the sum of all 20 of those numbers added together equals 1,410!)
Arithmetic vs. Geometric vs. Fibonacci
If you are looking at a worksheet and aren’t sure what kind of sequence you are dealing with, here is how to tell the primary types apart:
- Arithmetic Sequences: Relies entirely on addition or subtraction. The distance between every consecutive number is the same. Example: 5, 10, 15, 20… (The pattern is adding 5. This is the “Common Difference”).
- Geometric Sequences: Relies entirely on multiplication or division. Instead of adding a flat amount, the numbers scale exponentially. Example: 2, 6, 18, 54… (The pattern is multiplying by 3. This is the “Common Ratio”).
- Fibonacci Sequence: A special sequence (1, 1, 2, 3, 5, 8…) where each number is the sum of the two preceding ones. Because it does not have a single common difference or a single common ratio, it cannot be solved using standard Nth term formulas!
Common Sequence Formulas Reference Table
Some sequences in mathematics are so common that they have their own dedicated, simplified formulas. Use this quick-reference table to memorize the formulas for the most frequently tested sequences:
| Sequence Name | The Number Pattern | Nth Term Formula | Example (5th Term) |
| Even Numbers | 2, 4, 6, 8, 10… | 2n | 2(5) = 10 |
| Odd Numbers | 1, 3, 5, 7, 9… | 2n – 1 | 2(5) – 1 = 9 |
| Square Numbers | 1, 4, 9, 16, 25… | n^2 | 5^2 = 25 |
| Cube Numbers | 1, 8, 27, 64, 125… | n^3 | 5^3 = 125 |
| Triangular Numbers | 1, 3, 6, 10, 15… | n(n + 1) / 2 | 5(6) / 2 = 15 |
Frequently Asked Questions (FAQs)
What does ‘n’ stand for in math sequences?
The letter ‘n’ stands for the position (or index) of the term in the sequence. For example, if a sequence is 2, 4, 6, 8, then the 1st term (n=1) is 2, and the 4th term (n=4) is 8.
What if my sequence is decreasing?
If your sequence is going down continuously (e.g., 20, 15, 10, 5…), it is still an arithmetic sequence. However, your common difference is negative. In this example, you would simply enter “-5” into the calculator’s Common Difference box.
What does “Sum of First N Terms” mean?
This is also known as a Partial Sum or a Series. It simply means adding up all the numbers in the sequence up to your target point. If your sequence is 1, 2, 3, 4, 5, the Nth term for n=5 is simply 5. But the Sum of the first 5 terms is 15 (1 + 2 + 3 + 4 + 5 = 15).
Can a common ratio be a fraction or a decimal?
Yes! If your geometric sequence is getting smaller (e.g., 100, 50, 25, 12.5), you are dividing by 2 each time. In algebraic terms, this means you are multiplying by a half. You would enter 0.5 (or 1/2) as your Common Ratio.
Why is my geometric sequence alternating between positive and negative?
If a geometric sequence looks like 3, -6, 12, -24, 48… it means your Common Ratio is a negative number! In this case, you are multiplying by -2. When you multiply a positive by a negative, it becomes negative, and when you multiply a negative by a negative, it flips back to positive!