Name them with one letter: Consecutive integers are numbers next to each other, like 1, 2, 3.
Another example is 5, 6, 7.
Instead of using actual numbers, write them as:
n, n + 1, n + 2
This lets us prove something is true for every set of consecutive integers, not just one example.
Choose the version you find easiest: There is more than one correct way to write consecutive integers.
For example, three consecutive integers can be written as:
n, n + 1, n + 2
or
n + 1, n + 2, n + 3
or
n - 1, n, n + 1
All of these represent three consecutive integers.
Choose the version that makes the algebra easiest. For most questions, n, n + 1, n + 2 is the simplest choice.
IB-style question — sum of three consecutive integers
Prove that the sum of any three consecutive integers is a multiple of 3.
Step by step
- Write three consecutive integers.
- Add them.
- Remove the brackets and collect like terms.
- Take out a factor of 3.
Final answer
3(n+1) is 3 × a whole number, so the sum is a multiple of 3.
IB-style question — sum equals three times the middle integer
Prove that the sum of any three consecutive integers is three times the middle integer.
Step by step
- Write three consecutive integers centred on n.
- Add them.
- Remove the brackets and collect like terms.
Final answer
The middle integer is n. Since the sum equals 3n, it is exactly three times the middle integer.
Not convinced? Try actual numbers: Whenever a proof feels abstract, test it with simple numbers.
Take three consecutive integers:
1, 2, 3
Their sum is:
1 + 2 + 3 = 6
The middle integer is 2.
3 × 2 = 6
It works.
Try another set:
7, 8, 9
7 + 8 + 9 = 24
The middle integer is 8.
3 × 8 = 24
The proof shows this will happen for every set of three consecutive integers, not just these examples.
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Two useful proof ideas: (1) If you multiply two consecutive integers together, the answer is always even.
Example: 4 × 5 = 20 7 × 8 = 56
This works because one of the numbers must be even.
(2) Sometimes you need to prove something is never divisible by a number.
To do this, show there is always a little bit left over.
For example, if a number can always be written as:
3 × (a whole number) + 2
then dividing by 3 will always leave a remainder of 2, so it can never be a multiple of 3.
IB-style question — a product is even
Prove that the product of any two consecutive integers is even.
Step by step
- Write two consecutive integers and their product.
- Of any two consecutive integers, exactly one is even.
- Examples:
- A multiplication with an even number gives an even answer.
Final answer
One of the two consecutive integers is always even, so their product is always even.
IB-style question — never a multiple of 3
Prove that the sum of the squares of any three consecutive integers is never a multiple of 3.
Step by step
- Square and add the three integers.
- Expand each bracket.
- Collect like terms.
- Rewrite 5 as 3 + 2.
- Now factor out the 3 from the first four terms.
Final answer
The sum is always 3 × (a whole number) + 2, so dividing by 3 always leaves remainder 2. Therefore it can never be a multiple of 3.
A leftover means “never”: The +2 is the key idea.
Because there is always a leftover 2 after dividing by 3, the expression can never be exactly divisible by 3.
Example:
8 = 3×2 + 2 11 = 3×3 + 2 14 = 3×4 + 2
They all leave remainder 2 when divided by 3.
Whenever you end up with:
3 × (a whole number) + 2
or
5 × (a whole number) + 1
or any similar leftover, you have proved the number is never a multiple of that number.
Useful facts about consecutive integers: • Sum of 3 consecutive integers = 3 × the middle integer.
Example: 1 + 2 + 3 = 6 Middle integer = 2 3 × 2 = 6
• If you multiply any 2 consecutive integers together, the answer is always even.
Example: 4 × 5 = 20 (even) 7 × 8 = 56 (even)
This happens because one of the two numbers must be even.
• When a question asks you to prove something is divisible by a number, add or multiply the consecutive integers and look for that factor.