Principle of Mathematical Induction MCQ

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• What is the induction hypothesis assumption for the inequality m ! > 2m where m>=4?
  ক

  for m=k, k+1!>2k holds

  খ

  for m=k, k!>2k holds

  গ

  for m=k, k!>3k holds

  ঘ

  for m=k, k!>2k+1 holds
• Which of the following is the base case for 4n+1 > (n+1)2 where n = 2?
  ক

  64 > 9

  খ

  16 > 2

  গ

  27 < 91

  ঘ

  54 > 8
• According to principle of mathematical induction, if P(k+1) = m(k+1) + 5 is true then _____ must be true.
  ক

  P(k) = 3m(k)

  খ

  P(k) = m(k) + 5

  গ

  P(k) = m(k+2) + 5

  ঘ

  P(k) = m(k)
• For any positive integer m ______ is divisible by 4.
  ক

  5m2 + 2

  খ

  3m + 1

  গ

  m2 + 3

  ঘ

  m3 + 3m
• For every natural number k, which of the following is true?
  ক

  (mn)k = mknk

  খ

  m*k = n + 1

  গ

  (m+n)k = k + 1

  ঘ

  mkn = mnk
• For any integer m>=3, the series 2+4+6+…+(4m) can be equivalent to ________
  ক

  m2+3

  খ

  m+1

  গ

  mm

  ঘ

  3m2+4
• For m = 1, 2, …, 4m+2 is a multiple of ________
  ক

  3

  খ

  5

  গ

  6

  ঘ

  2
• In the principle of mathematical induction, which of the following steps is mandatory?
  ক

  induction hypothesis

  খ

  inductive reference

  গ

  induction set assumption

  ঘ

  minimal set representation
• What is the base case for the inequality 7n > n3, where n = 3?
  ক

  652 > 189

  খ

  42 < 132

  গ

  343 > 27

  ঘ

  42 <= 431
• What will be P(k + 1) for P(n) = n3 (n + 1)?
  ক

  (k + 1)4

  খ

  k4 + 5k3 + 9k2 + 7k + 2

  গ

  k4 + 6k3 + 9k2 + 7k + 2

  ঘ

  k4 + 3k3 + 9k2 + 6k + 2
• n3 + 5n is divisible by which of the following?
  ক

  3

  খ

  5

  গ

  7

  ঘ

  11
• n2 + 3n is always divisible by which number, provided n is an integer?
  ক

  2

  খ

  3

  গ

  4

  ঘ

  5
• If P(k) = k2 (k + 3) (k2 – 1) is true, then what is P(k + 1)?
  ক

  (k + 1)2 (k + 3) (k2 – 1)

  খ

  (k + 1)2 (k + 4) (k2 – 1)

  গ

  (k + 1)2 (k + 4) k (k + 2)

  ঘ

  (k + 1) (k + 4) k (k +2)
• What would be the hypothesis of mathematical induction for n(n + 1) < n! (where n ≥ 4) ?
  ক

  It is assumed that at n = k, k(k + 1)! > k!

  খ

  It is assumed that at n = k, k(k + 1)! < k!

  গ

  It is assumed that at n = k, k(k + 1)! > (k + 1)!

  ঘ

  It is assumed that at n = k, (k + 1)(k + 2)! < k!
• P(n) = n(n2 – 1). Which of the following does not divide P(k+1)?
  ক

  k

  খ

  k + 2

  গ

  k + 3

  ঘ

  k + 1
• If 103n + 24k + 1. 9 + k, is divisible by 11, then what is the least positive value of k?
  ক

  7

  খ

  6

  গ

  8

  ঘ

  10
• By principle of mathematical induction, 24n-1 is divisible by which of the following?
  ক

  8

  খ

  3

  গ

  5

  ঘ

  7
• 72n + 22n – 2. 3n – 1 is divisible by 50 by principle of mathematical induction.
  ক

  True

  খ

  False
• For principle of mathematical induction to be true, what type of number should ‘n’ be?
  ক

  Whole number

  খ

  Natural number

  গ

  Rational number

  ঘ

  Any form of number