What are the formulas for generic sequences

Question

What are the formulas for generic sequences

Answer 1

One formula for a generic sequence is the general term formula, also known as the nth term formula. This formula allows you to find the value of any term in the sequence directly, without having to calculate all the previous terms.

The general term formula can be written as:

an = f(n)

where an represents the value of the nth term in the sequence and f(n) is the function that describes how the terms are generated.

Another formula that can be used for generic sequences is the recursive formula. This formula defines each term in the sequence in terms of the previous terms.

The recursive formula can be written as:

a1 = c
an = g(an-1)

where a1 is the initial term of the sequence, c is a constant or initial value, and g(an-1) is a function that uses the previous term to generate the current term.

These formulas can be used to describe a wide range of generic sequences, allowing you to find specific terms or generate the sequence as a whole.

Answer 2

In mathematics, there are several formulas for different types of generic sequences. Let me provide you with some commonly used formulas:

1. Arithmetic sequence:
- The general formula for an arithmetic sequence is: 𝑎𝑛 = 𝑎₁ + (𝑛−1)𝑑.
- Here, 𝑎𝑛 represents the 𝑛th term of the sequence, 𝑎₁ is the first term, 𝑛 is the position of the term, and 𝑑 is the common difference between consecutive terms.

2. Geometric sequence:
- The general formula for a geometric sequence is: 𝑎𝑛 = 𝑎₁ × 𝑟^(𝑛−1).
- Here, 𝑎𝑛 represents the 𝑛th term of the sequence, 𝑎₁ is the first term, 𝑛 is the position of the term, and 𝑟 is the common ratio between consecutive terms.

3. Quadratic sequence:
- The general formula for a quadratic sequence is: 𝑎𝑛 = 𝑎 + 𝑏𝑛 + 𝑐𝑛².
- Here, 𝑎𝑛 represents the 𝑛th term of the sequence, 𝑎, 𝑏, and 𝑐 are constants, and 𝑛 is the position of the term.

4. Fibonacci sequence:
- The general formula for the 𝑛th term of the Fibonacci sequence is given by: 𝑛 = 𝑛₋₁ + 𝑛₋₂.
- Here, 𝑛₋₁ represents the (𝑛-1)th term and 𝑛₋₂ represents the (𝑛-2)th term of the Fibonacci sequence.

These are just a few examples. There are many other types of sequences, each with its own specific formula.