Mathematical Induction - Divisibility questions.
Test Yourself 1.
Prove the following relationships true for all natural numbers ≥ 1 except where indicated otherwise.
| One term with an index. | n3 - n is always divisble by 3. | n(n + 1)(n + 2) is divisible by 3. |
| 5n + 2×11n is a multiple of 3. | 5n - 1 is divisible by 4. | |
| 9n - 3 is divisible by 6. | 4n + 14 is a multiple of 6. | |
| (xn - 1) is divisible by (x - 1). | 5n + 12n - 1 is divisible by 16. | |
| Multiple terms with an index. | 5n + 2×11n is divisible by 3. | 23n - 3n is divisible by 5. |
(i) Given that f(k) = 12k + 2 × 5k-1, show that f(k+1) - 5f(k) = 1 × 12k where a is an integer. (ii) Hence or otherwise, prove by mathematical induction that 12n + 2×5n-1 is divisible by 7. |
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| n3 + 3n2 + 2n is divisible by 6. |
32n-1 + 5 is divisible by 8. | |
| 7n + 11n is divisible by 9. | 52n + 3n - 1 is divisible by 9. | |
| n3 + (n + 1)3 + (n + 2)3 is divisible by 9. | 3n + 7n is divisible by 10. | |
| 47n + 53 × 147n-1 is divisible by 100. | The sum of the cubes of three consecutive integers is divisible by 3. | |