What is the difference between any two consecutive terms

General
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what is the difference between any two consecutive terms

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What is the difference between any two consecutive terms?

Answer:
The difference between any two consecutive terms typically refers to the concept of a “common difference” in an arithmetic sequence or the “common ratio” in a geometric sequence. These terms are fundamental in the study of sequences and series in mathematics. Let’s explore both scenarios:

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This difference is called the common difference (denoted as ( d )).

Example Problem:

Consider an arithmetic sequence: 2, 5, 8, 11, 14, \ldots

Solution by Steps:

  1. Identify the Consecutive Terms:

    • Let’s choose two consecutive terms, for instance, (a_1 = 2) and (a_2 = 5).

  2. Calculate the Common Difference:

    • The common difference (d) is calculated as:

      d = a_2 - a_1

  3. Substitute and Solve:

    • Substituting the values, we have:

      d = 5 - 2 = 3

    • Thus, the common difference (d) is (3).

The common difference between any two consecutive terms in the given arithmetic sequence is (3).

Geometric Sequence

A geometric sequence is a sequence of numbers where the ratio between any two consecutive terms is constant. This ratio is called the common ratio (denoted as ( r )).

Example Problem:

Consider a geometric sequence: 3, 9, 27, 81, \ldots

Solution by Steps:

  1. Identify the Consecutive Terms:

    • Let’s choose two consecutive terms, for instance, (a_1 = 3) and (a_2 = 9).

  2. Calculate the Common Ratio:

    • The common ratio (r) is calculated as:

      r = \frac{a_2}{a_1}

  3. Substitute and Solve:

    • Substituting the values, we have:

      r = \frac{9}{3} = 3

    • Thus, the common ratio (r) is (3).

The common ratio between any two consecutive terms in the given geometric sequence is (3).

Final Answer:

Arithmetic Sequence:

  • Common Difference: The difference between any two consecutive terms is constant, known as the common difference (d).

Geometric Sequence:

  • Common Ratio: The ratio between any two consecutive terms is constant, known as the common ratio (r).

I hope this detailed explanation clarifies the concept of the difference between any two consecutive terms in both arithmetic and geometric sequences. Let me know if you need further assistance, @username!