Which formula can be used to describe the sequence

General
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which formula can be used to describe the sequence

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Which formula can be used to describe the sequence?

Answer:
Describing a sequence formula depends significantly on the type and nature of the sequence in question. Sequences can be arithmetic, geometric, or may follow a more complex or less regular pattern. I’ll provide an overview of how to derive formulas for different types of sequences.

1. Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the “common difference” and is denoted by d.

  • General Formula:

    If (a_n) represents the (n)-th term of an arithmetic sequence with the first term (a_1) and common difference (d), the (n)-th term is given by:

    a_n = a_1 + (n - 1)d

Example:

Consider the sequence (2, 5, 8, 11, \ldots). Here, the first term (a_1 = 2) and the common difference (d = 3).

Using the formula:

a_n = 2 + (n - 1) \cdot 3 = 2 + 3n - 3 = 3n - 1

Therefore, the formula describing this sequence is (a_n = 3n - 1).

2. Geometric Sequences

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the “common ratio” (r).

  • General Formula:

    If (a_n) represents the (n)-th term of a geometric sequence with the first term (a_1) and common ratio (r), the (n)-th term is given by:

    a_n = a_1 \cdot r^{n-1}

Example:

Consider the sequence (3, 6, 12, 24, \ldots). Here, the first term (a_1 = 3) and the common ratio (r = 2).

Using the formula:

a_n = 3 \cdot 2^{n-1}

Therefore, the formula describing this sequence is (a_n = 3 \cdot 2^{n-1}).

3. Quadratic Sequences

A quadratic sequence is one where the second differences between terms are constant.

  • General Formula:

    For a quadratic sequence, the (n)-th term is generally represented as:

    a_n = an^2 + bn + c

Here, (a), (b), and (c) are constants that can be determined using the initial terms of the sequence.

Example:

Consider the sequence (2, 6, 12, 20, \ldots).

  1. First term (a_1 = 2)
  2. Second term (a_2 = 6)
  3. Third term (a_3 = 12)

To find (a), (b), and (c):

a_1 = a \cdot 1^2 + b \cdot 1 + c = a + b + c = 2
a_2 = a \cdot 2^2 + b \cdot 2 + c = 4a + 2b + c = 6
a_3 = a \cdot 3^2 + b \cdot 3 + c = 9a + 3b + c = 12

By solving these simultaneous equations, we can determine the values of (a), (b), and (c).

  1. a + b + c = 2
  2. 4a + 2b + c = 6
  3. 9a + 3b + c = 12

Solving these,

a = 1, b = 1, c = 0

Thus, the general formula for this quadratic sequence is:

a_n = n^2 + n

4. Complex or Non-Linear Sequences

For sequences that do not follow a simple arithmetic, geometric, or quadratic pattern, more sophisticated methods or recurrence relations may be needed. These sequences often require more context or a deeper understanding of the specific problem to derive an appropriate formula.

Final Answer:

To determine the exact formula for a given sequence, the sequence type (arithmetic, geometric, quadratic, etc.) must be identified. Corresponding formulas are:

  • Arithmetic Sequence: a_n = a_1 + (n - 1)d
  • Geometric Sequence: a_n = a_1 \cdot r^{n-1}
  • Quadratic Sequence: a_n = an^2 + bn + c

For more complex sequences, additional context or information about the pattern is necessary to derive a particular formula.