Mathematical Induction Questions And Answers Pdf

Mathematical induction problems

Mathematical induction and Divisibility problems:

Ques. For all positive integral values of n, 3²ⁿ – 2n + 1 is divisible by
(a) 2
(b) 4
(c) 8
(d) 12
Ans. (a)

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Ques. If n ∈ N, then x^{2n – 1} + y^{2n – 1} is divisible by
(a) x + y
(b) x – y
(c) x² + y²
(d) x² + xy
Ans. (a)

Ques. If n ∈ N, then 7²ⁿ + 2^{3n – 3}. 3^{n – 1} is always divisible by
(a) 25
(b) 35
(c) 45
(d) None of these
Ans. (a)

Ques. If p is a prime number, then n^p – n is divisible by p when n is a
(a) Natural number greater than 1
(b) Irrational number
(c) Complex number
(d) Odd number
Ans. (a)

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Ques. For every natural number n, n(n + 1) is always
(a) Even
(b) Odd
(c) Multiple of 3
(d) Multiple of 4
Ans. (a)

Ques. If n ∈ N, then 11^{n + 2} + 12^{2n + 1} is divisible by
(a) 113
(b) 123
(c) 133
(d) None of these
Ans. (c)

Ques. For every natural number n,

n(n² – 1) is divisible by
(a) 4
(b) 6
(c) 10
(d) None of these
Ans. (b)

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Ques. The statement P(n) "1 x 1! + 2 x 2! + 3 x 3! + … + n x n! = (n + 1)! – 1" is
(a) True for all n > 1
(b) Not true for any n
(c) True for all n ∈ N
(d) None of these
Ans. (c)

Ques. For every natural number n
(a) n > 2ⁿ
(b) n < 2ⁿ(c) n = 2
(d) n = 2n²
Ans. (b)

Ques. The remainder when 5⁹⁹ is divided by 13 is
(a) 6
(b) 8
(c) 9
(d) 10
Ans. (b)

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Ques. For each n ∈ N, the correct statement is
(a) 2ⁿ < n
(b) n² > 2n
(c) n⁴ < nⁿ
(d) 2³ⁿ > 7n + 1
Ans. (c)

Ques. 10ⁿ + 3(4ⁿ⁺²) + 5 is divisible by (n ∈ N)
(a) 7
(b) 5
(c) 9
(d) 17
Ans. (c)

Ques. For natural number n, (n!)² > nⁿ, if
(a) n > 3
(b) n > 4
(c) n ³ 4
(d) n ³ 3
Ans. (d)

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Ques. For natural number n, 2ⁿ (n – 1)! < nⁿ, if
(a) n < 2
(b) n > 2
(c) n ³ 2
(d) Never
Ans. (b)

Ques. For positive integer n, 10^{n – 2} > 81n, if
(a) n > 5
(b) n ³ 5
(c) n < 5
(d) n > 6
Ans. (b)

Ques. When 2³⁰¹ is divided by 5, the least positive remainder is
(a) 4
(b) 8
(c) 2
(d) 6
Ans. (c)

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Ques. For every positive integer n, 2ⁿ < n! when
(a) n < 4
(b) n ³ 4
(c) n < 3
(d) None of these
Ans. (b)

Ques. x(x^n–1 – na^n–1) + aⁿ(n–1) is divisible by (x – a)² for
(a) n > 1
(b) n > 2
(c) All n ∈ N
(d) None of these
Ans. (c)

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Ques. For every positive integral value of n, 3ⁿ > n³ when
(a) n > 2
(b) n ³ 3
(c) n ³ 4
(d) n < 4
Ans. (c)