Mathematical induction and Divisibility problems:

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

Ans. (a)

Related: Electric field test questions

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)

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)

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)

Related: p Block elements questions

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

Ans. (a)

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)

For every natural number n, n(n² – 1) is divisible by
(a) 4
(b) 6
(c) 10
(d) None of these

Ans. (b)

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)

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

Ans. (b)

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

Ans. (b)

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

Ans. (c)

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

Ans. (c)

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