Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Year 8: expand 3(n + 4).
Answer
3n + 12. - From Year 9: n is a whole number. What is the next even number after 2n?
Answer
2n + 2. - From Year 9: is 2n + 1 even or odd when n is a whole number?
Answer
Odd. It is one more than an even number.
Words used in this lesson
- consecutive
- following one after another with no gaps. For example: 8, 9, 10.
- even
- a multiple of 2, written 2n. For example: 14.
- odd
- one more than a multiple of 2, written 2n + 1. For example: 27.
- multiple
- a number you get by multiplying by a whole number. For example: 5 × 7.
- proof
- an argument that a statement is true for every case. For example: 7(2n + 1) is a multiple of 7.
- expression
- numbers and letters with operations, but no equals sign. For example: 2n + 5.
Why does the total never surprise you?
A teacher in Dar es Salaam asks a class to pick any three consecutive whole numbers, add them, and write the total on the board. Everyone chooses a different starting number. The totals are different, but the teacher says that all of them have something in common.
Predict first: what could every total have in common? Try five different starting numbers on paper, and write down your guess. Then use the figure, which also lets the numbers be spread out by a gap d that is not 1.
n = 1 and d = 1: the numbers are 1, 2 and 3. Their total is 6 = 3 × 2, and 2 is the middle number.
Show me the steps
- n = 1, d = 1: the numbers are 1, 2, 3. The total is 6 = 3 × 2.
- n = 4, d = 1: the numbers are 4, 5, 6. The total is 15 = 3 × 5, three times the middle number.
- n = 4, d = 3: the numbers are 4, 7, 10. The total is 21 = 3 × 7. The gap is not 1, but the total is still three times the middle number.
- Any n and d: n + (n + d) + (n + 2d) = 3n + 3d = 3(n + d). The total is always three times the middle number n + d.
A letter stands for every case. If n is any whole number, then 2n is even, 2n + 1 is odd, and n, n + 1, n + 2 are consecutive. Write the statement with letters, expand, collect like terms, and look for a factor in front of a bracket.
A proof ends with a sentence: the expression is 2 × (something), so it is even for every whole n. Checking examples is only a test; the factor is the proof.
Two common mistakes. Using the same letter for two different numbers: two even numbers are 2a and 2b, not 2n and 2n, unless they are equal. And ending with a number: “it works for 5 and 8” is not a proof; the last line must show the factor, for example 3(n + 1).
Worked examples
1. Prove that the sum of three consecutive whole numbers is a multiple of 3.
- Call them n, n + 1 and n + 2. Their sum is n + (n + 1) + (n + 2) = 3n + 3.
- 3n + 3 = 3(n + 1), which is 3 times a whole number, so it is a multiple of 3 for every n.
2. Prove that the sum of two odd numbers is even.
- Two odd numbers are 2a + 1 and 2b + 1 (different letters, because they may be different). Sum: 2a + 2b + 2.
- 2a + 2b + 2 = 2(a + b + 1), which is 2 times a whole number, so it is even.
3. Prove that (n + 1)² − n² is always odd.
- Expand: (n + 1)² = n² + 2n + 1, so (n + 1)² − n² = 2n + 1.
- 2n + 1 is one more than 2n, an even number, so it is odd for every whole number n.
Your go. The first step is given. Prove that the product of an even number and any whole number is even. Let the even number be 2a and the other whole number be b.
- The product is 2a × b = 2ab.
- Now write 2ab as 2 times a whole number.
Check the answer
2ab = 2(ab). Since a and b are whole numbers, ab is a whole number, so the product is 2 times a whole number: it is even for every a and b.
Check before you go on
1. Three consecutive even numbers start at 2n. Which expression is their sum?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. Four consecutive whole numbers n, n + 1, n + 2 and n + 3 add to 4n + ?. What number goes in place of ?
3. Three consecutive odd numbers can be written 2n − 1, 2n + 1 and 2n + 3. They add to 6n + ?. What number goes in place of ?
Level 2 · Building
4. Five consecutive whole numbers have middle number m. They add to ?m. What number goes in place of ?
5. Two odd numbers are 2a + 1 and 2b + 1. Expand their product (2a + 1)(2b + 1) and write it as 2(2ab + a + b) + ?. What number goes in place of ?
Level 3 · Stretch
6. n² + n = n(n + 1) is a product of two consecutive whole numbers. For how many whole numbers n from 1 to 20 is n² + n even?
7. Expand and simplify (n + 3)² − (n − 3)². The result is ?n. What number goes in place of ?
Something to think about
The total of three consecutive whole numbers is always a multiple of 3. Is the total of six consecutive whole numbers always a multiple of 6? Test 1 to 6, then use a letter.
One way to think about it
No. 1 + 2 + 3 + 4 + 5 + 6 = 21, which is not a multiple of 6. In general n + (n + 1) + … + (n + 5) = 6n + 15 = 6(n + 2) + 3, which is 3 more than a multiple of 6 for every n. One example disproves the claim; the algebra says why it fails every time.
Exit ticket
8. Which statement is a proof that the sum of two consecutive odd numbers is a multiple of 4?