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Number Sequence Calculator

Result: —
Arithmetic Sequence:
Tₙ = a + (n - 1)d
Sₙ = n/2 × [2a + (n - 1)d]
Example:
a = 2, d = 5, n = 4 → T₄ = 2 + 3×5 = 17, S₄ = 4×(2 + 15)/2 = 34
Geometric Sequence:
Tₙ = a × rⁿ⁻¹
Sₙ = a(1 - rⁿ)/(1 - r), r ≠ 1
Example:
a = 3, r = 2, n = 4 → T₄ = 3×2³ = 24, S₄ = 3(1 - 2⁴)/(1 - 2) = 45
Fibonacci Sequence:
F(n) = F(n-1) + F(n-2)
Starts with 0, 1
Example:
F(7) = 0, 1, 1, 2, 3, 5, 8 → F₇ = 8
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A number sequence is an ordered list of numbers that follow a specific pattern or rule. Each number in the list is called a term, and the total number of terms is known as the length of the sequence. Sequences can be finite or extend infinitely.

In number sequences, the order matters, and it’s common for some values to repeat. There are many types of sequences in mathematics, but some of the most common include:

  • Arithmetic sequences

  • Geometric sequences

  • Fibonacci sequences

What Are Sequences Used For?

Sequences are widely used across various branches of mathematics, especially in studying patterns, limits, and functions. They play a key role in topics like:

  • Calculus and analysis

  • Differential equations

  • Series and convergence

If a sequence tends toward a specific value as it progresses, it’s called convergent. If not, it’s called divergent.

Notation in Sequences

Simple patterns may be listed directly (e.g., 2, 4, 6, 8…), but more complex sequences often use index notation to define the general term based on its position in the sequence.

Arithmetic Sequence

An arithmetic sequence is a set of numbers where the difference between terms stays the same. This difference is called the common difference and can be either positive or negative.

Arithmetic Sequence Formula:

General term:

aₙ = a₁ + d × (n - 1)

Where:

  • aₙ = nth term

  • a₁ = first term

  • d = common difference

  • n = term number

Example:

Sequence: 1, 3, 5, 7, 9, 11...

Common difference d = 2

To find the 5th term:

a₅ = 1 + 2 × (5 - 1) = 1 + 8 = 9

Sum of an Arithmetic Sequence:

To find the sum of the first n terms:

Sum = (n × (a₁ + aₙ)) / 2

Example:

 
1 + 3 + 5 + 7 + 9 = 25
Sum = (5 × (1 + 9)) / 2 = 25

Geometric Sequence

A geometric sequence is a sequence in which each term after the first is found by multiplying the previous term by a fixed number, called the common ratio.

Geometric Sequence Formula:

General term:

aₙ = a × rⁿ⁻¹

Where:

  • a = first term (scale factor)

  • r = common ratio

  • n = term number

  • aₙ = nth term

Example:

Sequence: 1, 2, 4, 8, 16, 32...

Here, a = 1, r = 2. To find the 8th term:

a₈ = 1 × 2⁷ = 128

Sum of a Geometric Sequence:

For the first n terms of a geometric sequence where r ≠ 1:

Sum = a × (1 - rⁿ) / (1 - r)

Example (first 3 terms):

1 + 2 + 4 = 7
Sum = 1 × (1 - 2³) / (1 - 2) = -7 / -1 = 7

Fibonacci Sequence

The Fibonacci sequence is a unique number series where each term is the sum of the two previous terms. It typically starts with 0 and 1 or 1 and 1, depending on the chosen definition.

Fibonacci Formula:

aₙ = aₙ₋₁ + aₙ₋₂

Where:

  • aₙ = nth term

  • aₙ₋₁ = previous term

  • aₙ₋₂ = term before that

Example:

Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21...

The first two terms are:

a₀ = 0
a₁ = 1

Fibonacci Applications

The Fibonacci sequence appears in various real-world settings such as:

  • Nature (e.g., branching of trees, flower petals)

  • Computer science (e.g., recursion algorithms)

  • Financial modeling

  • Art and architecture (e.g., golden ratio relationships)

Conclusion

This Number Sequence Calculator is a helpful tool for identifying patterns, calculating terms, and finding sums in different types of sequences. Whether you’re working with arithmetic, geometric, or Fibonacci sequences, understanding these foundational concepts will enhance your grasp of mathematical patterns and problem-solving techniques.

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