Sequence and Series Introduction
10 Questions
2 Views

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to lesson

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What is the smallest number expressible as the sum of two cubes in two different ways?

1729

What is a sequence?

A function whose domain is the set N of natural numbers.

Which of the following is a type of sequence?

  • Finite sequences
  • Infinite sequences
  • Both A and B (correct)
  • None of the above
  • If a sequence has a finite number of terms, it is called a ______ sequence.

    <p>finite</p> Signup and view all the answers

    What formula is used to find the nth term of an arithmetic progression (A.P.)?

    <p>tn = a + (n - 1)d</p> Signup and view all the answers

    In the arithmetic progression 4, 7, 10, 13,..., what is the common difference?

    <p>3</p> Signup and view all the answers

    What is the sum of the first n terms of an A.P. represented as?

    <p>Sn = n/2 * [2a + (n - 1)d]</p> Signup and view all the answers

    How many terms are there in the sequence 4, 7, 10, 13,...,82?

    <p>27</p> Signup and view all the answers

    In an A.P., what does the rth term formula express?

    <p>tr = Sr - Sr-1</p> Signup and view all the answers

    What is the first term 'a' and common difference 'd' in the A.P. where the 3rd term is 18 and the 7th term is 30?

    <p>a = 12, d = 3</p> Signup and view all the answers

    Study Notes

    Sequence

    • A sequence is defined as a function mapping natural numbers (N) to real numbers (R).
    • Denoted as f(n) = tn, where n is a natural number. A sequence is represented by its range: {t1, t2, t3, ...}.

    Real Sequence

    • A real sequence has values that are a subset of real numbers (R).
    • Examples include:
      • (i) 2, 5, 8, 11, ...
      • (ii) 4, 1, -2, -5, ...

    Types of Sequence

    • Finite Sequences: Contain a limited number of terms.
    • Infinite Sequences: Extend indefinitely with an infinite number of terms.
    • Example: For tn = (\frac{(-2)^n}{(-1)^n + 2}), specific terms yield a sequence of values.

    Series

    • A series is formed by the addition or subtraction of a sequence's terms.
    • If a1, a2, ..., an are terms of a sequence, then the series is expressed as: a1 + a2 + ... + an.
    • Examples of series include:
      • (i) 1 + 2 + 3 + ... + n
      • (ii) 2 + 4 + 8 + 16 + ...
      • (iii) -1 + 3 - 9 + 27 - ...

    Progression

    • The term "progression" refers broadly to sequences or series, both finite and infinite.

    Arithmetic Progression (A.P.)

    • An A.P. consists of terms generated by adding a constant number 'd' (common difference) to the previous term.
    • A.P. representation: a, a + d, a + 2d, ..., a + (n - 1)d.
    • Example: Sequence -4, -1, 2, 5.

    nth Term of an A.P.

    • The nth term in an A.P. can be calculated using the formula: (tn = a + (n - 1)d).
    • Here, 'a' is the first term and 'd' is the common difference.

    Sum of First n Terms of an A.P.

    • The formula for the sum of the first n terms (Sn) of an A.P. is:
      • ( S_n = \frac{n}{2} [2a + (n - 1)d] )
      • For n odd: ( S_n = [a + \lambda] t_{(n + 1)/2} ) where λ is the last term.

    Example Calculations

    • Finding the number of terms in A.P. (4, 7, 10, ..., 82):

      • Use (82 = 4 + (n - 1)3) to determine that n = 27.
    • For an A.P. where the 3rd term is 18 and the 7th term is 30:

      • Establish relations to find a and d, yielding the sum ( S_{17} = 612 ).

    Sum of Odd Numbers Between 1 and 1000 Divisible by 3

    • Odd numbers sequence within the range includes 3, 5, 7, ..., 999; filtering for multiples of 3 results in a subset of terms.

    Studying That Suits You

    Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

    Quiz Team

    Description

    Explore the fascinating world of sequences and series in this quiz. Learn about their definitions, properties, and the significant contributions of mathematicians like S. Ramanujan. Test your understanding and deepen your knowledge of this essential mathematical concept.

    More Like This

    Number Sequence Quiz
    5 questions

    Number Sequence Quiz

    AmpleCarolingianArt avatar
    AmpleCarolingianArt
    Basic Number Sequence Quiz
    0 questions
    Arithmetic Progression (A.P) Basics
    9 questions
    Use Quizgecko on...
    Browser
    Browser