Number Sequence Calculator
Generate arithmetic and geometric sequences with our free number sequence calculator. Find nth term, sum, and visualize terms with interactive charts for any progression.
About This Calculator
Our free Number Sequence Calculator helps you generate arithmetic and geometric sequences instantly. Whether you are a student studying progressions, a teacher preparing classroom examples, or a professional analyzing data patterns, this calculator provides complete results including the full sequence, sum of terms, and the nth term for any arithmetic or geometric progression. Simply select the sequence type, enter the first term, common difference (for arithmetic) or common ratio (for geometric), and the number of terms you want.
For arithmetic sequences, the nth term is calculated using the formula an = a1 + (n-1)d, where a1 is the first term, d is the common difference, and n is the term number. The sum of the first n terms is Sn = n/2 × (2a1 + (n-1)d). For geometric sequences, the nth term is an = a1 × r(n-1), where r is the common ratio, and the sum is Sn = a1 × (1 - rn) / (1 - r) when r ≠ 1.
The calculator provides two visualization modes: a line chart showing the growth pattern of the sequence and a bar chart comparing each individual term. Results update instantly and URLs are shareable, making it easy to save or share specific sequence calculations with classmates or colleagues.
Applications
Arithmetic sequences model linear relationships such as fixed monthly deposits, constant speed travel, and uniform temperature changes. Geometric sequences model exponential phenomena including compound interest, population growth, inflation, and technological adoption rates. Understanding these patterns is essential in mathematics, finance, physics, biology, and economics.
Frequently Asked Questions
What is a number sequence?
A number sequence is an ordered list of numbers that follow a specific pattern or rule. Each number in the sequence is called a term. Common types include arithmetic sequences (constant difference between terms) and geometric sequences (constant ratio between terms).
What is the difference between an arithmetic and geometric sequence?
An arithmetic sequence has a constant difference between consecutive terms (e.g., 2, 4, 6, 8 has common difference 2). A geometric sequence has a constant ratio between consecutive terms (e.g., 2, 4, 8, 16 has common ratio 2). Arithmetic sequences grow linearly while geometric sequences grow exponentially.
How do I find the nth term of an arithmetic sequence?
The nth term of an arithmetic sequence is given by the formula an = a1 + (n-1)d, where a1 is the first term, d is the common difference, and n is the term number. For example, in the sequence 3, 7, 11, 15... (a1=3, d=4), the 10th term would be 3 + (10-1)×4 = 39.
How do I find the nth term of a geometric sequence?
The nth term of a geometric sequence is given by the formula an = a1 × r(n-1), where a1 is the first term, r is the common ratio, and n is the term number. For example, in the sequence 2, 6, 18, 54... (a1=2, r=3), the 5th term would be 2 × 3(5-1) = 162.
How do I calculate the sum of an arithmetic series?
The sum of the first n terms of an arithmetic series is Sn = n/2 × (2a1 + (n-1)d), where a1 is the first term and d is the common difference. Alternatively, you can use Sn = n/2 × (a1 + an), where an is the last term.
How do I calculate the sum of a geometric series?
The sum of the first n terms of a geometric series is Sn = a1 × (1 - rn) / (1 - r) when r ≠ 1. If r = 1, the sum is simply a1 × n. For example, the sum of the first 4 terms of 2, 6, 18, 54 is 2 × (1 - 34) / (1 - 3) = 80.
Where are number sequences used in real life?
Number sequences appear in many real-world contexts. Arithmetic sequences model linear growth like monthly savings, salary increments, and distance traveled at constant acceleration. Geometric sequences model exponential growth like population growth, compound interest, radioactive decay, and viral spread.
What is the common ratio in a geometric sequence?
The common ratio (r) in a geometric sequence is the constant factor by which each term is multiplied to get the next term. It is found by dividing any term by the previous term. For example, in the sequence 5, 10, 20, 40..., the common ratio is 10/5 = 20/10 = 40/20 = 2.