Number Sequence Calculator
Calculate terms and sums for Arithmetic, Geometric, and Fibonacci sequences
Calculate terms and sums for Arithmetic, Geometric, and Fibonacci sequences with detailed step-by-step explanations.
Arithmetic Sequence Calculator
an = a₁ + f × (n-1)
Example: 1, 3, 5, 7, 9, 11, 13, ...
Geometric Sequence Calculator
an = a × rⁿ⁻¹
Example: 1, 2, 4, 8, 16, 32, 64, 128, ...
Fibonacci Sequence Calculator
a₀=0; a₁=1; an = an-1 + an-2
Example: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
General Explanation of Sequences
A sequence is an ordered list of objects or numbers that follow a specific pattern. In mathematics, sequences are fundamental concepts used to represent ordered collections of elements. Each element in a sequence is called a "term," and the number of terms is called the "length" of the sequence.
Common types of sequences include arithmetic sequences, geometric sequences, and the Fibonacci sequence. Sequences have applications in various areas of mathematics, including convergence and divergence analysis, function theory, series, and differential equations. They can be represented in different ways: by listing the terms, using a general formula, or through indexing notation.
Arithmetic Sequence
An arithmetic sequence is a sequence where the difference between successive terms is constant. This constant difference is called the "common difference."
General Formulations:
an = a₁ + f × (n-1) (where an is the nth term)
an = am + f × (n-m) (where a₁ is the first term)
Notation Example:
i.e. a₁, a₁ + f, a₁ + 2f, ... f is the common difference
Example Sequence:
EX: 1, 3, 5, 7, 9, 11, 13, ...
Calculation Example (5th term):
Using the formula: an = a₁ + f × (n-1)
a₅ = 1 + 2 × (5-1) = 1 + 2 × 4 = 1 + 8 = 9
Result: 9
Sum Formula:
n × (a₁ + an) / 2
Sum Calculation Example (through 5th term):
Sum: 1+3+5+7+9 = 25
Using formula: (5 × (1+9)) / 2 = 50/2 = 25
Result: 25
Geometric Sequence
A geometric sequence is a sequence where each successive number is a multiplication of the previous number by a fixed, non-zero common ratio.
General Formulation:
an = a × rⁿ⁻¹ (where an is the nth term, a is the scale factor, r is the common ratio)
Notation Example:
i.e. a, ar, ar², ar³, ... a is the scale factor and r is the common ratio
Example Sequence:
EX: 1, 2, 4, 8, 16, 32, 64, 128, ...
Calculation Example (8th term):
Using the formula: an = a × rⁿ⁻¹
a₈ = 1 × 2⁷ = 1 × 128 = 128
Result: 128
Sum Formula:
a × (1-rⁿ) / (1-r)
Sum Calculation Example (through 3rd term):
Sum: 1+2+4 = 7
Using formula: 1 × (1-2³) / (1-2) = -7 / -1 = 7
Result: 7
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