Principle of Mathematical Induction MATHS - Principle of Mathematical Induction

For all n ∈ N, 3 × 5²ⁿ⁺¹ + 2³ⁿ⁺¹ is divisible by:

1. 19
2. 17
3. 23
4. 25

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

1. 5
2. 3
3. 7
4. 1

If xⁿ - 1 is divisible by x - k, then the least positive integral value of k is:

1. 1
2. 2
3. 3
4. 4

Justify: Let P(n) be a statement and let P(k) ⇒ P(k + 1), for some natural number k, then P(n) is true for all n ∈ N.

Prove the following statement by principle of mathematical induction:
For any natural number n, 7ⁿ - 2ⁿ is divisible by 5.

Prove the following statement by principle of mathematical induction:
2 + 4 + 6 + ..... + 2n = n² + n for all natural numbers n.

Prove the following statement by principle of mathematical induction:
2³ⁿ - 1 is divisible by 7, for all natural numbers n.

Prove the following statement by principle of mathematical induction:
3²ⁿ - 1 is divisible by 8, for all natural numbers n.