NCEA Level 1Generalising properties of numbersAS 91947

Generalising: 7 exam questions solved step by step

What this lesson covers

  • An example shows a statement can be true; only algebra shows it is always true.
  • Even numbers can be written 2n, odd numbers 2n + 1, and consecutive numbers n, n + 1, n + 2.
Key idea

7 questions, one method

  • 1. Define each number with a letter.
  • 2. Simplify, then factorise.
  • 3. Conclude in a sentence.
  • To disprove: one counterexample.
Your turn

Question 1 · Achieved

  • Prove that the sum of any two
  • even numbers is always even.
Worked example

Q1: even + even

  • Two numbers, two letters.
  • Factorise out the 2.
  1. Let m and n be integers
  2. evens: 2m and 2ntwo letters
  3. 2n + 2none number, twice
  4. 2m + 2nadd
  5. = 2(m + n)common factor 2
  6. m + n is an integer
  7. 2 × integer, so always even
Your turn

Question 2 · Achieved

  • Claim: for every positive whole
  • number n, n² + n + 1 is prime.
  • Show that the claim is false.
Worked example

Q2: is n² + n + 1 always prime?

  • Try values in order.
  • One failure is enough.
  1. n = 1: 1 + 1 + 1 = 3prime
  2. n = 2: 4 + 2 + 1 = 7prime
  3. n = 3: 9 + 3 + 1 = 13prime
  4. n = 4: 16 + 4 + 1 = 21
  5. 21 = 3 × 7not prime
  6. n = 4 is a counterexample
  7. The claim is false.
Your turn

Question 3 · Merit

  • Prove that the sum of any three
  • consecutive odd numbers is
  • always a multiple of 3.
Worked example

Q3: three consecutive odds

  • Odd numbers go up in 2s.
  • Target form: 3( ).
  1. Let n be an integer
  2. 2n + 1, 2n + 3, 2n + 5odds go up in 2s
  3. (2n + 1) + (2n + 3) + (2n + 5)
  4. = 6n + 9collect like terms
  5. = 3(2n + 3)take out the 3
  6. 2n + 3 is an integer
  7. 3 × integer: a multiple of 3
  8. 1 + 3 + 5 = 9 = 3 × 3
Your turn

Question 4 · Merit, in context

  • On a calendar, pick a 2 × 2 square.
  • Top right × bottom left, minus
  • top left × bottom right.
  • Prove the answer is always 7.
One 2 × 2 square of dates
WedThu
910
1617
Worked example

Q4: the calendar square

  • Next day +1, next week +7.
  • Bracket what you subtract.
  1. top: n and n + 1next day is +1
  2. bottom: n + 7 and n + 8a week later is +7
  3. (n + 1)(n + 7) − n(n + 8)the difference
  4. (n + 1)(n + 7) = n² + 8n + 7
  5. n(n + 8) = n² + 8n
  6. n² + 8n + 7 − (n² + 8n)keep the bracket
  7. = 7n² and 8n cancel
  8. always 7, for every square
  9. 10 × 16 − 9 × 17 = 160 − 153 = 7
Your turn

Question 5 · Merit, in context

  • Think of an amount in dollars.
  • Add $3, multiply by 4, subtract $8.
  • Divide by 4, take away your amount.
  • Prove you always end with $1.
Worked example

Q5: the money trick

  • One example is not proof.
  • One line per instruction.
  1. $6, 9, 36, 28, 7, then 7 − 6 = $1
  2. Let the amount be $n
  3. n + 3add 3
  4. 4(n + 3) = 4n + 12multiply by 4
  5. 4n + 12 − 8 = 4n + 4subtract 8
  6. (4n + 4) ÷ 4 = n + 1divide every term
  7. n + 1 − n = 1take away n
  8. always $1, for any amount
Your turn

Question 6 · Excellence

  • Prove: the difference between the
  • squares of two consecutive odd
  • numbers is always a multiple of 8.
Worked example

Q6: form and expand

  • Larger square first.
  • Square in full: 3 terms.
  1. Let n be an integer
  2. odds: 2n − 1 and 2n + 1
  3. (2n + 1)² − (2n − 1)²larger minus smaller
  4. (2n + 1)² = 4n² + 4n + 1
  5. (2n + 1)² = 4n² + 1middle term missing
  6. (2n − 1)² = 4n² − 4n + 1
Worked example

Q6: subtract and conclude

  • Minus before a bracket:
  • every sign inside changes.
  1. (2n + 1)² − (2n − 1)²
  2. 4n² + 4n + 1 − (4n² − 4n + 1)
  3. = 4n² + 4n + 1 − 4n² + 4n − 1
  4. = 8n4n² and 1 cancel
  5. 8 × integer: a multiple of 8
  6. 7² − 5² = 49 − 25 = 24 = 8 × 3
Your turn

Question 7 · Excellence

  • Mia: the sum of any three
  • consecutive even numbers is
  • always a multiple of 12.
  • Is she right? Prove what is true.
Worked example

Q7: a multiple of 12?

  • Prove what is true.
  • Disprove what is not.
  1. 2n, 2n + 2, 2n + 4n an integer, up in 2s
  2. 2n + (2n + 2) + (2n + 4)add them
  3. = 6n + 6collect
  4. = 6(n + 1)common factor 6
  5. n + 1 integer: a multiple of 6
  6. × 12 only if n + 1 is even
  7. 4 + 6 + 8 = 18not a multiple of 12
  8. 2 + 4 + 6 = 12one example fools you
  9. Wrong: a multiple of 6, not 12
Summary

The method that settled all seven

  • Define, simplify, factorise, conclude, or one counterexample.
StepHow
Defineeven 2n, odd 2n + 1
Letterstwo numbers, two letters
Simplifya minus hits every term
Factorise2( ), 2( ) + 1 or k( )
Concludename the integer, restate
Disproveone counterexample

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