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Q1. If a statement is to be proved by mathematical induction, then the different steps necessary to prove it are

• Basic step to be proved
• Inductive step to be proved
• Prove P(1), P(2), P(3)
• Prove Basic step and Inductive step

Solution

The principle of mathematical induction is a tool which can be used to prove a wide variety of mathematical statements. Each such statement   is  assumed as P(n) associated with positive integer n, for which the correctness for the case n = 1 is examined which is basic step. Then assuming the truth of P(k) for some positive integer k, the truth of P(k + 1) is established which is inductive step,Then by  principle of mathematical induction the statement  P(n) is  true for all n  N.

Q2. Every even power of an odd number greater than 1 when divided by 8 leaves 1 as the remainder.The inductive step for the above statement is

• P(1) = 3² = 8.1 + 1
• P(k) = (2k - 1)² = 8a + 1 implies P(k + 1); (2k + 1)² = 8b + 1 for some natural  numbers a,b
• P(k) = (2k + 1)² = 8a + 1 implies P(k + 1):(2k + 3)² = 8b + 1 for some natural numbers a,b
• P(k) = (2k + 1)²ⁿ = 8a + 1 implies P(k + 1) : (2k + 3)^{2n + 2} = 8b + 1 for some natural number a,b

Solution

The statement in mathematical form is P(n) = (2n + 1)²ⁿ = 8a + 1 for some natural number a The Basic Step is P(1) : 3² = 8.1 + 1Inductive Step is If P(k) is true then P(k + 1) also is truei.e P(k) = (2k + 1)²ⁿ = 8a + 1 Let us prove this is true for n = k + 1   Therefore, (2k + 3)^{2n + 2} = 8A + 1 for some natural numbers A.

Q3. 2.4^{2n + 1} + 3^{3n + 1} for all n is divisible by

• 2
• 3
• 4
• 11

Solution

For n = 1, 2.4^{2n + 1} + 3^{3n + 1} = 2.4³ + 3⁴ = 128 + 81 = 209 which is divisible by 11.

Q4. If P(k) is the statement 2^3k - 1 is divisible by 7, then P(k + 1) is

• 2^3k+1 - 1 is divisible by 7
• 2^{3k - 2} is divisible by 7
• 2^3k+2 - 1 is divisible by 7
• 2^{3k +3} - 1 is divisible by 7

Solution

P(k): 2^3k - 1 is divisible by 7. P(k + 1): 2^{3k + 3}- 1 is divisible by 7.

Q5. The sum of cubes of three consecutive natural numbers is divisible by:

• 2
• 5
• 7
• 9

Solution

Q6. If P(n) : n³ + n is divisible by 3, then which of the following is true

• P (1) is true
• P (2) is true
• P (3) is true
• P (4) is true

Solution

P(3) = 3³ + 3 = 27 + 3 = 30 is divisible by3.

Q7. The greatest natural number, which divides (n + 1) (n +2) (n + 3)(n + 4) is

• 1
• 4
• 24
• 6

Solution

Since product of r consecutive integers is divisible by r!. (n + 1)(n + 2)(n + 3)(n + 4) is divisible by 4! = 24

Q8. _______ reasoning depends on working with each case, and developing a conjecture by observing incidence till each and every case is observed.

• Inductive Reasoning
• Deductive Reasoning
• Mathematical Reasoning
• Logical Reasoning

Solution

Inductive reasoning depends on working with each case, and developing a conjecture by observing incidence till each and every  case is observed.

Q9. P(n): n²+n+1 is prime for n=1,2.......40. The least number for which the result does not hold is

• 41
• 42
• 51
• 47

Solution

P(n): n² + n + 1P(41): 41² + 41 + 1 = 1723 which is a prime numberP(42): 42² + 42 + 1= 18071807 is divisible by 13 and hence not prime.

Q10. The principle of mathematical induction is for the set of:

• Whole number
• Integers
• Positive integers
• Rational numbers

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