Sequences and Series in Mathematics
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Questions and Answers

In a Fibonacci sequence, what is the value of $a_5$ given that $a_1 = 1$, $a_2 = 1$?

  • 2
  • 3
  • 8 (correct)
  • 5
  • Which of the following is a correct representation of a Geometric Progression (G.P.)?

  • 1, 3, 6, 10, 15
  • 2, 5, 8, 11, 14
  • 1, 2, 4, 8, 16
  • 9, 27, 81, 243 (correct)
  • If the nth term of a sequence is defined by the formula $a_n = a_{n-1} - 1$ for $n > 2$, and if $a_2 = 2$, what is the value of $a_4$?

  • 3
  • 1
  • 2
  • 0 (correct)
  • In a sequence where $a_1 = -1$ and the recursive formula is given by $a_n = rac{a_{n-1}}{n}$ for $n ≥ 2$, what is the value of $a_3$?

    <p>-0.25</p> Signup and view all the answers

    Which statement correctly describes the relationship between the Fibonacci sequence and the growth of natural numbers?

    <p>Fibonacci sequence describes the recursive relationship of natural growth.</p> Signup and view all the answers

    Study Notes

    Introduction

    • Natural numbers are a product of human spirit – Dedekind
    • Sequences can be used to represent patterns and relationships in data
    • Series are used to represent the sum of a sequence's terms

    Types of Sequences and Series

    • Arithmetic Progression (A.P.)
      • Defined by a common difference between consecutive terms
      • Example: 2, 4, 6, 8...where the common difference is 2
    • Geometric Progression (G.P.)
      • Common ratio between consecutive terms
      • Example: 2, 4, 8, 16… where the common ratio is 2

    Fibonacci Sequence

    • Defined by the sum of the previous two terms
    • First two terms are 1, 1
    • Example: 1, 1, 2, 3, 5, 8, 13…
    • Formula: an = an–1 + an–2, n > 2
    • Golden ratio (phi): approximately 1.618
      • The ratio of two consecutive Fibonacci numbers approaches phi as the numbers increase

    Geometric Progression (G.P.)

    • Example: 2, 4, 8, 16…
    • Each term is found by multiplying the previous term by a common ratio
      • (In this case, the common ratio is 2)
    • Examples:
      • 2, 4, 8, 16…
      • 1/9, 1/27, 1/81, 1/243
      • .01, .0001, .000001…

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    Sequence and Series PDF

    Description

    Explore the fundamental concepts of sequences and series, including arithmetic and geometric progressions, as well as the famous Fibonacci sequence. This quiz covers definitions, examples, and patterns that define these important mathematical concepts.

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