Math Sequences and Series
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Questions and Answers

What is the primary difference between a sequence and a series?

  • A sequence is finite, while a series is infinite.
  • A sequence has a fixed constant, while a series has a variable constant.
  • A sequence is arithmetic, while a series is geometric.
  • A sequence is the list of numbers, while a series is the sum of those numbers. (correct)

    • What is the formula for the sum of an arithmetic series?

  • Σa_n = (n/2)(a_1 + a_n) (correct)
  • Σa_n = (n/2)(a_1 - a_n)
  • Σa_n = a_1 / (1 - r), |r| < 1
  • Σa_n = a_1(1 - r^n) / (1 - r)

    • What is the definition of a harmonic sequence?

  • A sequence in which each term is the reciprocal of an arithmetic sequence. (correct)
  • A sequence in which each term is obtained by adding a fixed constant to the previous term.
  • A sequence in which each term is the sum of the previous two terms.
  • A sequence in which each term is obtained by multiplying the previous term by a fixed constant.

    • What is the condition for a geometric series to be convergent?

    <p>|r| &lt; 1</p> Signup and view all the answers

    What is the formula for the sum of an infinite geometric series?

    <p>Σa_n = a_1 / (1 - r), |r| &lt; 1</p> Signup and view all the answers

    What type of sequence is the series 1 + 1/2 + 1/4 + 1/8 + ...?

    <p>Geometric sequence</p> Signup and view all the answers

    Study Notes

    Sequence and Series

    Sequences

    • A sequence is an ordered list of numbers, denoted by {a_n} where n is a positive integer.
    • Each term in the sequence is denoted by a_n, where n is the term number.
    • A sequence can be finite (has a last term) or infinite (has no last term).

    Types of Sequences

    • Arithmetic sequence: a sequence in which each term is obtained by adding a fixed constant to the previous term.
      • Example: 2, 5, 8, 11, ...
    • Geometric sequence: a sequence in which each term is obtained by multiplying the previous term by a fixed constant.
      • Example: 2, 6, 18, 34, ...
    • Harmonic sequence: a sequence in which each term is the reciprocal of an arithmetic sequence.
      • Example: 1, 1/2, 1/3, 1/4, ...

    Series

    • A series is the sum of the terms of a sequence.
    • The sum of a series can be finite or infinite.
    • A series can be represented as:
      • Σa_n = a_1 + a_2 + ... + a_n

    Types of Series

    • Arithmetic series: the sum of an arithmetic sequence.
      • Example: 2 + 5 + 8 + 11 + ...
    • Geometric series: the sum of a geometric sequence.
      • Example: 2 + 6 + 18 + 34 + ...
    • Convergent series: a series that has a finite sum.
      • Example: 1 + 1/2 + 1/4 + 1/8 + ...
    • Divergent series: a series that has an infinite sum.
      • Example: 1 + 2 + 4 + 8 + ...

    Formulas

    • Arithmetic series formula: Σa_n = (n/2)(a_1 + a_n)
    • Geometric series formula: Σa_n = a_1(1 - r^n) / (1 - r)
    • Sum of an infinite geometric series: Σa_n = a_1 / (1 - r), |r| < 1

    Sequences

    • A sequence is an ordered list of numbers, denoted by {a_n}, where n is a positive integer.
    • Each term in the sequence is denoted by a_n, where n is the term number.
    • Sequences can be finite or infinite.

    Types of Sequences

    • Arithmetic sequence: each term is obtained by adding a fixed constant to the previous term.
    • Geometric sequence: each term is obtained by multiplying the previous term by a fixed constant.
    • Harmonic sequence: each term is the reciprocal of an arithmetic sequence.

    Series

    • A series is the sum of the terms of a sequence.
    • The sum of a series can be finite or infinite.
    • A series can be represented as: Σa_n = a_1 + a_2 +...+ a_n.

    Types of Series

    • Arithmetic series: the sum of an arithmetic sequence.
    • Geometric series: the sum of a geometric sequence.
    • Convergent series: a series that has a finite sum.
    • Divergent series: a series that has an infinite sum.

    Formulas

    • Arithmetic series formula: Σa_n = (n/2)(a_1 + a_n).
    • Geometric series formula: Σa_n = a_1(1 - r^n) / (1 - r).
    • Sum of an infinite geometric series: Σa_n = a_1 / (1 - r), |r| < 1.

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    Description

    This quiz covers the basics of sequences and series, including types of sequences such as arithmetic and geometric sequences. Test your understanding of these mathematical concepts!

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