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Quiz

Year 3: Mathematical induction

10 questions with solutions for year 3: identities, inequalities and divisibility proved by induction – checked for particular n.

Type the number or choose the right answer. When you finish, click Check answers and see the solutions.

1. The first step of a proof by induction is:

2. 2 + 4 + 6 + … + 2n = n(n + 1). What is 2 + 4 + … + 20?

3. 2² + 4² + … + (2n)² = (2/3)n(n + 1)(2n + 1). What is 2² + 4² + 6²?

4. 1/(1·4) + 1/(4·7) + 1/(7·10) = ?

5. 1/(1·5) + 1/(5·9) = ?

6. 2 + 16 + 56 + … + (3n − 2)·2ⁿ = 10 + (3n − 5)·2ⁿ⁺¹. What is the sum for n = 3?

7. 3ⁿ > n⁴ holds for n > 7. What is the smallest natural n for which it holds?

8. n³ > 3n + 3 holds for n > 2. What is the smallest natural n for which it holds?

9. 7 | 2ⁿ⁺² + 3²ⁿ⁺¹. What is 2ⁿ⁺² + 3²ⁿ⁺¹ for n = 1?

10. 17 | 3·5²ⁿ⁺¹ + 2³ⁿ⁺¹. For n = 1, what is the quotient on dividing by 17?