RR RR RRR ORR flashcards

All 12 cards

1. Card 1

   What defines a sequence of real numbers?

   A sequence of real numbers is a function (rule) whose domain is the set of all natural numbers and whose range is a subset of R. It is essentially a function on the set of positive integers.

2. Card 2

   Explain the difference between a finite and an infinite sequence.

   A sequence with a finite number of terms is called a finite sequence. If a sequence does not have a finite number of terms, it is considered an infinite sequence.

3. Card 3

   What is an arithmetic sequence (A.P.)?

   An arithmetic sequence is a sequence where there is a uniform common difference between consecutive terms. This means that each term after the first is obtained by adding a constant value to the preceding term.

4. Card 4

   If the mth term of an A.P. is n and the nth term is m, what is the rth term?

   • A. m-n-r
   • B. m+n-r (correct)
   • C. m-n+r
   • D. n-r-m

5. Card 5

   What is the formula for the sum of the first n terms of an A.P. (S_n)?

   The sum of the first n terms of an A.P. is given by the formula S_n = n/2 * [2a + (n-1)d], where 'a' is the first term, 'd' is the common difference, and 'n' is the number of terms.

6. Card 6

   If a, b, c are in A.P., then 3^a, 3^b, 3^c are in which type of sequence?

   • A. A.P.
   • B. G.P. (correct)
   • C. H.P.
   • D. None of these

7. Card 7

   Define a geometric sequence (G.P.) and its nth term.

   A geometric sequence is a sequence where there is a uniform common ratio between any two consecutive terms. If 'a' is the first term and 'r' is the common ratio, the nth term of the series is t_n = ar^(n-1).

8. Card 8

   What is the sum to infinity of a geometric series?

   The sum to infinity of a geometric series with first term 'a' and common ratio 'r' (where |r| < 1) is given by the formula S_infinity = a / (1-r).

9. Card 9

   If a, b, c are three unequal numbers such that a, b, c are in A.P. and (b-a), (c-b), a are in G.P., what is the ratio a:b:c?

   • A. 2:3:4
   • B. 1:2:3 (correct)
   • C. 3:4:5
   • D. 4:5:6

10. Card 10

    How is a Harmonic Series or Sequence defined?

    A harmonic series is formed by taking the reciprocal of each term of an arithmetic series. If a_1, a_2, a_3, ... are in A.P., then 1/a_1, 1/a_2, 1/a_3, ... are in H.P.

11. Card 11

    What is the relationship between Arithmetic Mean (A.M.), Geometric Mean (G.M.), and Harmonic Mean (H.M.) for two unequal positive numbers?

    For any two unequal positive numbers, the Arithmetic Mean (A.M.), Geometric Mean (G.M.), and Harmonic Mean (H.M.) satisfy the relationship A.M. > G.M. > H.M. Additionally, A.M. * H.M. = (G.M.)^2.

12. Card 12

    If x, y, z are in G.P., then logx, logy, logz are in which type of sequence?

    • A. A.P. (correct)
    • B. G.P.
    • C. H.P.
    • D. None of these

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