1 Marks Question

Question 11 Mark

State the first principle of mathematical induction.

Answer

Let p(n) be a statement involving the natural number n such that:
p(1) is true
p(m + 1) is true, whenever p(m) is true
Then p(n) is true for all $\text{n}\in\text{N}$
This is called first principle of Mathematical Induction.

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Question 21 Mark

Write the set of value if n for which the statement p(n): 2n < n! is true.

Answer

N - {1, 2, 3} where N is the set of all natural numbers.

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Question 31 Mark

State the second principle of mathematical induction.

Answer

Let p(n) be a statement involving the natural number n such that:
p(1) is true
p(m + 1) is true, whenever p(m) is true for all $\text{n}\leq\text{m}$
Then p(n) is true for all $\text{n}\in\text{N}$
This is called first principle of Mathematical Induction.

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Question 41 Mark

If p(n): 2 × 4^2n-1 + 3³ⁿ⁺¹ is divisible by $\lambda$ for all $\text{n}\in\text{N}$ is true, then find the value of $\lambda$

Answer

2 × 42ⁿ⁺¹+ 33ⁿ⁺¹
= 2 × 24ⁿ⁺² + 3³ⁿ⁺¹
= 2⁴ⁿ⁺³ + 3³ⁿ⁺¹
Given above expression is divisible by $\lambda$ for all $\text{n}\in\text{N}$
So lets check for n = 1
For n = 2
= 2048 + 2187 = 4235
Now its clearly evident that common factor for above numbers is 11 so $\lambda = 11$

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Question 51 Mark

If P(n) is the statement "n(n + 1) is even", then what is P(3)?

Answer

P(n): n(n + 1) is even.
P(3): 3(3 + 1) is even.

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Question 61 Mark

Given an example of a statement P(n) such that it is true for all $\text{n}\in\text{N}.$

Answer

P(n): $1 + 2 + 3 + ... + \text{n}=\frac{\text{n}(\text{n}+1)}{2}$ is true for all $\text{n}\in\text{N}.$

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