Sequences in Mathematics
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Questions and Answers

What is a sequence in mathematics?

  • An unordered list of numbers
  • A sum of numbers
  • A product of numbers
  • An ordered list of numbers (correct)

    • What is the general term of an arithmetic sequence?

  • a_n = a_(n-1) + a_(n-2)
  • a_n = 1/n
  • a_n = a_1 \* r^(n-1)
  • a_n = a_1 + (n-1)d (correct)

    • What is the sum of a geometric sequence called?

  • Geometric Series (correct)
  • Arithmetic Series
  • Harmonic Series
  • Fibonacci Series

    • What is the general term of a harmonic sequence?

    <p>a_n = 1/n</p> Signup and view all the answers

    What is the Fibonacci sequence?

    <p>A sequence of numbers in which each term is the sum of the two preceding terms</p> Signup and view all the answers

    What is a convergent series?

    <p>A series with a finite sum as the number of terms increases without bound</p> Signup and view all the answers

    What is the general term of a geometric series?

    <p>S_n = a_1 * (1 - r^n) / (1 - r)</p> Signup and view all the answers

    What is the sum of an arithmetic sequence called?

    <p>Arithmetic Series</p> Signup and view all the answers

    Study Notes

    Sequence

    • A sequence is an ordered list of numbers, often denoted as {a_n} or (a_n)
    • Each term in the sequence is called an element or a term
    • A sequence can be finite (having a fixed number of terms) or infinite (having an infinite number of terms)
    • A sequence can be represented recursively or explicitly

    Types of Sequences

    • Arithmetic Sequence (AP):
      • Each term is obtained by adding a fixed constant to the previous term
      • General term: a_n = a_1 + (n-1)d, where d is the common difference
    • Geometric Sequence (GP):
      • Each term is obtained by multiplying the previous term by a fixed constant
      • General term: a_n = a_1 * r^(n-1), where r is the common ratio
    • Harmonic Sequence:
      • A sequence of reciprocals of positive integers
      • General term: a_n = 1/n
    • Fibonacci Sequence:
      • A sequence of numbers in which each term is the sum of the two preceding terms
      • General term: a_n = a_(n-1) + a_(n-2), with a_1 = a_2 = 1

    Series

    • A series is the sum of the terms of a sequence
    • A series can be finite or infinite
    • A series can be denoted as Σa_n or ∑a_n

    Types of Series

    • Arithmetic Series:
      • The sum of an arithmetic sequence
      • General term: S_n = n/2 * (a_1 + a_n)
    • Geometric Series:
      • The sum of a geometric sequence
      • General term: S_n = a_1 * (1 - r^n) / (1 - r), where |r| < 1
    • Harmonic Series:
      • The sum of a harmonic sequence
      • General term: S_n = 1 + 1/2 + 1/3 + ... + 1/n
    • Convergent and Divergent Series:
      • A convergent series has a finite sum as the number of terms increases without bound
      • A divergent series has an infinite sum or no sum at all

    Sequence

    • An ordered list of numbers, denoted as {a_n} or (a_n)
    • Each term is an element or a term in the sequence
    • Sequences can be finite or infinite
    • Sequences can be represented recursively or explicitly

    Types of Sequences

    Arithmetic Sequence (AP)

    • Each term is obtained by adding a fixed constant to the previous term
    • General term: a_n = a_1 + (n-1)d, where d is the common difference

    Geometric Sequence (GP)

    • Each term is obtained by multiplying the previous term by a fixed constant
    • General term: a_n = a_1 * r^(n-1), where r is the common ratio

    Harmonic Sequence

    • A sequence of reciprocals of positive integers
    • General term: a_n = 1/n

    Fibonacci Sequence

    • A sequence of numbers where each term is the sum of the two preceding terms
    • General term: a_n = a_(n-1) + a_(n-2), with a_1 = a_2 = 1

    Series

    • The sum of the terms of a sequence
    • A series can be finite or infinite
    • A series can be denoted as Σa_n or ∑a_n

    Types of Series

    Arithmetic Series

    • The sum of an arithmetic sequence
    • General term: S_n = n/2 * (a_1 + a_n)

    Geometric Series

    • The sum of a geometric sequence
    • General term: S_n = a_1 * (1 - r^n) / (1 - r), where |r| < 1

    Harmonic Series

    • The sum of a harmonic sequence
    • General term: S_n = 1 + 1/2 + 1/3 +...+ 1/n

    Convergent and Divergent Series

    • A convergent series has a finite sum as the number of terms increases without bound
    • A divergent series has an infinite sum or no sum at all

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    Description

    Learn about sequences, including its definition, types, and representation. Understand finite and infinite sequences, and explore arithmetic sequences and their general terms.

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