Sequence
Generate arithmetic and geometric sequences instantly. Enter first term, common difference or ratio, and number of terms to get the sequence, sum, and nth term with formulas.
About This Calculator
A sequence is an ordered list of numbers that follow a specific pattern or rule. The two most common types are arithmetic sequences (constant difference between consecutive terms) and geometric sequences (constant ratio between consecutive terms). Understanding sequences is fundamental to algebra, calculus, and countless real-world applications from finance to physics.
This sequence calculator generates both arithmetic and geometric sequences. For arithmetic sequences, it uses the formula aₙ = a + (n-1)d where a is the first term and d is the common difference. For geometric sequences, it uses aₙ = a x rⁿ⁻¹ where r is the common ratio. The calculator also computes the sum of the first n terms using the standard formulas: Sₙ = n/2 x (2a + (n-1)d) for arithmetic and Sₙ = a(1-rⁿ)/(1-r) for geometric sequences.
The results include the full comma-separated sequence, the sum of all generated terms, and the value of the nth term. This tool is ideal for students studying sequences, professionals analyzing growth patterns, or anyone working with numerical patterns in data analysis, financial projections, or mathematical modeling.
Regional Notes
Sequences are taught globally in secondary and higher-secondary mathematics. In India, arithmetic and geometric progressions are covered in CBSE Class 10 and Class 11. In the US, sequences appear in Algebra 2 and Precalculus courses. In the UK, sequences are part of GCSE and A-Level mathematics curricula. The formulas and properties are universal across all educational systems.
Frequently Asked Questions
What is an arithmetic sequence?
An arithmetic sequence is a sequence where each term differs from the previous term by a constant value called the common difference (d). For example, 3, 7, 11, 15, 19 is an arithmetic sequence with first term a=3 and common difference d=4. The nth term is given by formula aₙ = a + (n-1)d.
What is a geometric sequence?
A geometric sequence is a sequence where each term is multiplied by a constant value called the common ratio (r) to get the next term. For example, 3, 6, 12, 24, 48 is a geometric sequence with first term a=3 and common ratio r=2. The nth term is given by formula aₙ = a x rⁿ⁻¹.
How do you calculate the sum of an arithmetic sequence?
The sum of the first n terms of an arithmetic sequence is Sₙ = n/2 x (2a + (n-1)d), where a is the first term and d is the common difference. Alternatively, Sₙ = n/2 x (a + l), where l is the last term. This calculator uses the standard formula for both arithmetic and geometric sequences.
How do you calculate the sum of a geometric sequence?
The sum of the first n terms of a geometric sequence is Sₙ = a(1-rⁿ)/(1-r) when r ≠ 1, where a is the first term and r is the common ratio. When r = 1, the sum is simply Sₙ = a x n. This calculator handles both cases automatically with the correct formula.
What are real-world applications of sequences?
Arithmetic sequences model linear growth such as monthly savings, annual salary increments, or rental increases. Geometric sequences model exponential growth and decay including population growth, compound interest, radioactive decay, viral spread, and loan amortization. Sequences are fundamental in calculus, finance, and data science.
Can the common difference or ratio be negative?
Yes, both common difference (arithmetic) and common ratio (geometric) can be negative. In arithmetic sequences, a negative difference produces a decreasing sequence. In geometric sequences, a negative ratio causes alternating signs between positive and negative terms. Our calculator handles all real number inputs for both parameters.
What is the difference between a sequence and a series?
A sequence is an ordered list of numbers following a pattern (e.g., 2, 4, 6, 8, 10), while a series is the sum of the terms of a sequence (e.g., 2+4+6+8+10 = 30). This calculator generates the sequence and also computes the sum of the first n terms, showing both the sequence values and the total.