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

Given. P(n) = n(n + 1) is even.

Find. P(3) ?

If P(n) is the statement “n^{3} + n is divisible by 3”, prove that P(3) is true but P(4) is not true.

Given. P(n) = n^{3} + n is divisible by 3

Find P(3) is true but P(4) is not true

If P(n) is the statement “2^{n} ≥ 3n”, and if P(r) is true, prove that P(r + 1) is true.

Given. P(n) = “2^{n} ≥ 3n” and p(r) is true.

Prove. P(r + 1) is true

If P(n) is the statement “n^{2} + n” is even”, and if P(r) is true, then P(r + 1) is true

Given. P(n) = n^{2} + n is even and P(r) is true.

Given an example of a statement P(n) such that it is true for all n ϵ N.

If P(n) is the statement “n^{2} – n + 41 is prime”, prove that P(1), P(2) and P(3) are true. Prove also that P(41) is not true.

Given. P(n) = n^{2} - n + 41 is prime

Prove. P(1),P(2) and P(3) are true and P(41) is not true.

Prove the following by the principle of mathematical induction:

i.e., the sum of the first n natural numbers is

To prove: Prove that by the Mathematical Induction.

i.e., the sum of first n odd natural numbers is n^{2}.

1.2 + 2.2^{2} + 3.2^{3} + … + n.2^{n}=(n–1) 2^{n + 1} + 2

2 + 5 + 8 + 11 + … + (3n – 1) = 1/2 n(3n + 1)

1.3 + 2.4 + 3.5 + … + n . (n + 2)

1.3 + 3.5 + 5.7 + … + (2n – 1) (2n + 1)

1.2 + 2.3 + 3.4 + … + n(n + 1)

1^{2} + 3^{2} + 5^{2} + … + (2n – 1)^{2}

a + ar + ar^{2} + … + ar^{n – 1}

a + (a + d) + (a + 2d) + … + (a + (n– 1)d)

5^{2n} – 1 is divisible by 24 for all n ϵ N

3^{2n} + 7 is divisible by 8 for all n ϵ N

5^{2n + 2} – 24n – 25 is divisible by 576 for all n ϵ N.

3^{2n + 2} – 8n – 9 is divisible by 8 for all n ϵ N.

(ab)^{n} = a^{n} b^{n} for all n ϵ N

Show that: (ab)^{n} = a^{n} b^{n} for all n ϵ N by Mathematical Induction

n(n + 1) (n + 5) is a multiple of 3 for alln ϵ N.

Show that: P(n): n(n + 1) (n + 5) is multiple by 3 for all n∈N

7^{2n} + 2^{3n – 3} . 3n – 1 is divisible by 25 for all n ϵ N

2.7^{n} + 3.5^{n} – 5 is divisible by 24 for all n ϵ N

11^{n+2} + 12^{2n+1} is divisible by 133 for all n ϵ N

1×1! + 2×2! + 3×3! +…+ n×n! = (n + 1)! – 1 for all n ϵ N.

n^{3} – 7n + 3 is divisible by 3 for all n ϵ N.

1 + 2 + 2^{2} + … + 2^{n} = 2^{n + 1} – 1 for all n ϵ N

Prove that 7 + 77 + 777 + … + 777 for all n ϵ N

Prove that is a positive integer for all n ϵ N

Prove that for all n ϵ N and

Prove that for all natural

numbers, n ≥ 2.

Prove that for all n ϵ N

Prove that for all n > 2, n ϵ N.

Prove that x^{2n – 1} + y^{2n – 1} is divisible by x + y for all n ϵ N.

Prove that sin x + sin 3x + … + sin (2n – 1) x for all

nϵN.

Prove that cos α + cos (α + β) + cos (α + 2β) + … + cos (α + (n – 1)β) for all n ϵ N

Prove that for all natural numbers n > 1.

Given and for n ≥ 2, where a > 0, A > 0.

Prove that

Let P(n) be the statement: 2^{n} ≥ 3n. If P(r) is true, show that P(r + 1) is true. Do you conclude that P(n) is true for all nϵN?

Show by the Principle of Mathematical induction that the sum S_{n} of the n terms of the series is given by

Prove that the number of subsets of a set containing n distinctelements is 2^{n} for all n ϵ N.

A sequence a_{1}, a_{2}, a_{3}, …... is defined by letting a_{1} = 3 and a_{k} = 7 a_{k – 1} for all natural numbers k ≥ 2. Show that a_{n} = 3.7 ^{n-1} for all n ϵ N

A sequence x_{1}, x_{2}, x_{3}, …. is defined by letting x_{1} = 2 and for all natural numbers k, k ≥ 2. Show that for all nϵN

A sequence x_{0}, x_{1}, x_{2}, x_{3}, …. is defined by letting x_{0} = 5 and

x_{k} =4+x_{k–1} for all natural numbers k. Show that x_{n} = 5 for all

nϵN using mathematical induction.

Using principle of mathematical induction prove that

for all natural numbers n ≥ 2.

The distributive law from algebra states that for real numbers

c, a_{1} and a_{2}, we have c(a_{1} + a_{2}) = c a_{1} + ca_{2}

Use this law and mathematical induction to prove that, for all

natural numbers, n ≥ 2, if c, a_{1}, a_{2}, …... a_{n} are any real numbers,

then c(a_{1} + a_{2} +…+ a_{n}) = c a_{1} + c a_{2} +…+ c a_{n}.

State the first principle of mathematical induction.

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

State the second principle of mathematical induction.

If P(n): 2 × 4^{2n + 1} + 3^{3n + 1} is divisible by λ for all n ∈ N is true, then find the value of λ.

Mark the Correct alternative in the following:

If x^{n} – 1 is divisible by x - λ, then the least positive integral value of λ is

For all n ∈ N, 3 × 5^{2n + 1} + 2^{3n + 1} is divisible by

If 10^{n} + 3 × 4^{n + 2} + λ is divisible by 9 for all n ∈ N, then the least positive integral value of λ is

Let P (n): 2n < (1 × 2 × 3 × … × n). Then the smallest positive integer for which P(n) is true is

A student was asked to prove a statement P(n) by induction. He proved P (k + 1) is true whenever P(k) is true for all k > 5 ∈ N and also P(5) is true. On the basis of this he could conclude that P(n) is true.

If P(n) : 49^{n} + 16^{n} + λ is divisible by 64 for n ∈ N is true, then the least negative integral value of λ is