Patterns are the hidden language of the universe. Whether you are looking at the growth of a bank account, the arrangement of petals on a flower, or a complex coding algorithm, sequences define the structure of our world. But identifying the "nth term" formula can be a mathematical headache.
Sequence Type: Arithmetic
Based on your input, the common difference is 2.
How to Use the Sequence Formula Calculator
To find the general rule for your sequence, simply enter the first few terms into the input field above. Separate each number with a comma. For the most accurate results, please provide at least 4 terms. Our calculator analyzes the differences and ratios between terms to determine if the sequence is:
- Arithmetic: Where the difference between consecutive terms is constant.
- Geometric: Where the ratio between consecutive terms is constant.
- Quadratic: Where the second difference between terms is constant.
Understanding the Different Types of Sequences
1. Arithmetic Sequences
An arithmetic sequence is the simplest form of progression. You add (or subtract) the same value to get from one term to the next. This constant value is called the "common difference" (d).
The standard formula is: an = a1 + (n - 1)d
For example, in the sequence 5, 8, 11, 14, the common difference is 3. The formula becomes an = 3n + 2.
2. Geometric Sequences
In a geometric sequence, you multiply by a constant value to get the next term. This value is the "common ratio" (r).
The standard formula is: an = a1 · r(n-1)
Consider 2, 6, 18, 54. Here, the common ratio is 3. The nth term is 2 · 3n-1.
3. Quadratic Sequences
When the first difference is not constant, but the difference of the differences is, you have a quadratic sequence. These follow the form: an = an2 + bn + c.
These are common in physics (projectile motion) and geometry (area calculations). Finding these manually involves solving a system of linear equations, but our calculator handles the heavy lifting for you.
Why Finding the Formula Matters
In data science and finance, being able to predict the next term in a sequence allows for forecasting. If you know the formula for your savings growth or your business expenses, you can project your financial status years into the future. It transforms a list of numbers into a predictable roadmap.