Proof by Deduction and Exhaustion · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Small certain steps, every case checked

Mathematics · Mathematical Thinking · ages 17-18
Name ______________________   Date ____________

  1. Write the 10th odd positive integer in the form 2n + 1 (with n starting at 0). What value do you get?

    Answer: ______________

  2. Which expression always gives an odd number when n is an integer?

    • 2n
    • 2n + 2
    • n squared + n
    • 2n + 1

  3. Which expression always gives an odd number when n is an integer?

    • 2n + 1
    • 2n
    • n squared + n

  4. Mo claims worked examples for n = 1, 2 and 3 prove a pattern for all n. Is Mo right?

    Circle one:   True   False

  5. What is (2 times 3 + 1) + (2 times 4 + 1)?

    Answer: ______________

  6. To prove a claim about the integers 1 to 5 by exhaustion, you must do what?

    • give a single example
    • find a general formula
    • check each of the 5 cases

  7. Two odd numbers 2n + 1 and 2m + 1 are added. What is their sum?

    • 2(n + m) + 1
    • 2(n + m + 1)
    • 2nm + 2
    • (n + m + 1) squared

  8. Since (2n + 1) + (2m + 1) = 2(n + m + 1), the sum of two odds is even. True or false?

    Circle one:   True   False

  9. Since (2n + 1) + (2m + 1) = 2(n + m + 1), the sum of two odds is even.

    Circle one:   True   False

  10. Why does checking every integer 1 to 5 prove a claim stated for those integers?

    • five examples prove any claim
    • examples beat algebra
    • every case the claim covers is checked
    • large numbers behave like small ones
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Small certain steps, every case checked W1-mt_QNWvbkg04f-s1

  1. 19 · 19. With n = 9, 2 times 9 + 1 = 19.
  2. 2n + 1 · 2n + 1. Doubling is always even, and one more than even is odd.
  3. 2n + 1 · Doubling is always even, and one more than even is odd.
  4. False · Examples illustrate; only a general argument proves for all n.
  5. 16 · 16. The brackets give 7 and 9, and 7 + 9 = 16.
  6. check each of the 5 cases · Exhaustion means no covered case goes unverified.
  7. 2(n + m + 1) · 2(n + m + 1). Adding gives 2n + 2m + 2, which factors as twice an integer: hence even.
  8. True · True. The sum is twice the integer n + m + 1, meeting the definition of even.
  9. True · The sum is twice the integer n + m + 1, meeting the definition of even.
  10. every case the claim covers is checked · Every case the claim covers is checked. With nothing left out, no counterexample can hide.
Worksheet · LightMySky