Proof by induction has two parts: the base case and the inductive step, in which you assume truth for n = k and prove it for n = k + 1, so this IB sheet lays out the machinery. The base case for positive integers is n = 1 in the answers.
Key Takeaways
Induction has the base case and the inductive step (Q1).
The inductive step moves from n = k to n = k + 1 (Q2).
Q3 records n = 1 as the base case.
Section A: Multiple Choice Questions (1 Mark Each)
Choose the correct option for each question.
1.Proof by induction has two parts: the base case and
(a) the inductive step(b) the final answer(c) a counter-example(d) a diagram
2.In the inductive step, you assume the statement is true for n = k and then prove it for
(a) n = k + 1(b) n = k − 1(c) n = 2k(d) n = k²
3.The base case for proving a statement for all positive integers usually starts at
(a) n = 1(b) n = 0 only(c) n = −1(d) n = 100
4.Proof by contradiction assumes
(a) the opposite of the statement and shows it leads to a contradiction(b) the statement is true(c) nothing at all(d) a diagram
5.To disprove a universal statement, it is enough to find
(a) one counterexample(b) two examples(c) a diagram(d) no evidence
6.In a direct proof, you
(a) use logical steps from known facts to reach the statement(b) assume the opposite(c) guess the answer(d) draw a picture
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.State the two parts of proof by induction.
8.What is the base case?
9.What do you assume in the inductive step?
10.What is proof by contradiction?
11.What is a counterexample?
12.Give one example of a statement that could be proved by induction.
13.What is the difference between a theorem and a proof?
14.What does it mean to prove a statement for all positive integers n?
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.The sum of the first n odd numbers is n². Check the base case n = 1.
16.For the formula S(n) = n(n + 1) ÷ 2, check the base case n = 1.
17.Assume a statement is true for n = k. What value of n do you prove it for in the inductive step?
18.Verify that n² − 1 is divisible by 8 for the base case n = 3.
19.The sum 1 + 2 + 3 + ... + n has the formula n(n + 1) ÷ 2. Evaluate it for n = 4.
20.Verify the base case for the statement 2ⁿ > n when n = 1.
Answer Key
1. a) the inductive step
2. a) n = k + 1
3. a) n = 1
4. a) the opposite of the statement and shows it leads to a contradiction
5. a) one counterexample
6. a) use logical steps from known facts to reach the statement
7. Refer to solution guide.
8. Refer to solution guide.
9. Refer to solution guide.
10. Refer to solution guide.
11. Refer to solution guide.
12. Refer to solution guide.
13. Refer to solution guide.
14. Refer to solution guide.
15. Refer to solution guide.
16. Refer to solution guide.
17. Refer to solution guide.
18. Refer to solution guide.
19. Refer to solution guide.
20. Refer to solution guide.
End of Worksheet
People Also Ask
What are the two parts of proof by induction?
The base case and the inductive step, option (a) in this worksheet (Q1).
What do you prove in the inductive step?
The statement for n = k + 1, option (a) in this worksheet (Q2).
What is the base case for positive integers?
n = 1, option (a) in this worksheet (Q3).
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