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This page sets out the method for proof in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Proof topic page.
When you see 'Prove that … is always …', do this.
When you see 'Is this statement true? Explain' or 'Show that the statement is not true', do this.
When you see 'Prove that is positive for all ', do this.
Let the first integer be n, or let an odd number be 2k+1. The letter stands for every case.
The next integer is n+1. The next odd number after 2k+1 is 2k+3.
Multiply the brackets before you factor. Do not replace the letter with 1.
A multiple of 4 should show 4 as a factor. 4(k+1) makes that obvious.
A^2-B^2=(A-B)(A+B). That identity is the whole proof for many square differences.
Checking n=1 does not prove a statement about every n. Your last line must show the factor, such as 4(k+1), for every value of the letter.
Consecutive integers share one letter: n and n+1. Their sum is 2n+1, which is one more than a multiple of 2, so it is odd. Their product is n(n+1). Among any two consecutive integers, one is even, so the product is a multiple of 2. You can also say: if n=2k the product is 2k(2k+1), and if n=2k+1 the product is (2k+1)(2k+2)=2(2k+1)(k+1). Either way a factor of 2 is on show.
Two odd numbers that are not the same, and not tied together, need two letters. 2n+1 and 2m+1. Using 2n+1 twice forces the two odds to be equal, and a proof about "any two odd numbers" has quietly become a proof about one of them. The sum (2n+1)+(2m+1)=2(n+m+1) is even. The product is odd, because it is 4nm+2n+2m+1=2(2nm+n+m)+1. The +1 at the end is what makes it odd.
Method: 1. Choose n for any integer before you expand. 2. Consecutive integers are n and n+1. Consecutive evens are 2n and 2n+2. 3. Expand and simplify until a factor is obvious. 4. End with a sentence that names the factor and says it holds for every integer n.
Consecutive odd numbers differ by 2. Write them as 2n+1 and 2n+3. Their sum is 4n+4=4(n+1), a multiple of 4. Check with n=5: the odds are 11 and 13, the sum is 24, and 4(5+1)=24. The algebra and the number agree. The sentence at the end is "the sum is 4(n+1), and n+1 is an integer, so the sum is a multiple of 4".
The same pair of odds has a difference of squares that is a multiple of 8. (2n+1)^2-(2n-1)^2. Expand both squares: 4n^2+4n+1 and 4n^2-4n+1. Subtract: (4n^2+4n+1)-(4n^2-4n+1)=8n. The 4n^2 and the +1 cancel, and you must change the signs of the whole second square. What remains is 8n, which is 8 times an integer. For n=3 the odds are 7 and 5, and 49-25=24=8× 3.
Method: 1. Odd numbers are 2n+1, 2n+3, and so on. 2. Add or multiply those expressions. Do not use particular odds such as 3 and 5 as the proof. 3. Factorise the result. 4. The factor you were asked for should be visible, with a sentence.
Stopping at 8n+8 is not finished. Factorise it as 8(n+1) and then say that this is a multiple of 8 because n+1 is an integer. The factor and the sentence are the last mark. The same applies to an even number written as 2(ldots) or an odd number written as 2(ldots)+1. Leave the expression in that shape. Do not expand it again afterwards.
A "show that" proof is the same work with the target written at the top. For (n+1)^2-(n-1)^2, expand to n^2+2n+1-(n^2-2n+1)=4n. The target is a multiple of 4, and 4n is exactly that. With n=7, 64-36=28 and 4× 7=28. If your expansion does not match the target, do not adjust the target. Find the sign error in the second bracket.
Method: 1. A line of algebra is not finished until you say what it shows. 2. Name the multiple: "which is a multiple of 4". 3. The sentence must match the factor you actually found. 4. If the factor is 8, do not claim only 4 unless 4 was the target and 8 implies it.
The claim "every odd number is one more than a multiple of 4" is false. Trying 1, 5 and 9 seems to support it, because 1=4(0)+1, 5=4(1)+1 and 9=4(2)+1. But 3 is odd and 3=4(0)+3 is not one more than a multiple of 4, so 3 is a counter-example. One counter-example kills the claim, and saying "it works" after three good examples proves nothing.
If a claim is true for every integer, no list of examples is long enough. The marker is looking for letters, an expansion, a factor, and a sentence. Numerical checks belong in the margin as a safety check on your expansion, the way 11+13=24 checks 4(n+1). They are not a substitute for the letters. Write them after the proof if you want, not instead of it.
Method: 1. Trying n=1 and n=2 can support a check, and can disprove a false claim. 2. Examples do not prove a statement about every integer. 3. One counter-example is enough to show a statement is false. 4. A proof uses a letter and covers every integer at once.
Every proof begins by naming the numbers. An even number is 2n, an odd number is 2n+1, and a multiple of 3 is 3n. The next consecutive even number after 2n is 2n+2 and the next odd number after 2n+1 is 2n+3. Writing 2n and 2n+1 for two consecutive even numbers is the most frequent error, so say the pattern aloud before you write it.
Exam questions often ask for the building block as its own early part, such as write down an expression for the next odd number after 2n+1. That mark is easy to collect and it sets up the proof that follows. When two numbers are not linked, use two different letters, for example 2m and 2n+1, so you do not force them to be equal.
Method: 1. Decide what kind of number you need: even, odd or a multiple of some number. 2. Write each number using a letter, adding 2 to move to the next of the same kind. 3. Use different letters for numbers that are not connected. 4. Keep the letters, expand and simplify before you try to conclude.
Tip: Say what n stands for. Write n is an integer at the start of your proof.
Some questions say two numbers differ by a given amount, they are multiplied, and then a number is added. The aim is to show the answer is always a square. If the integers are n and n+8, the product is n(n+8)=n^2+8n. Adding 16 gives n^2+8n+16, and this factorises as (n+4)^2, which is a square because n+4 is an integer.
The pattern is that the number added is the square of half the difference. A difference of 6 needs 9, and a difference of 8 needs 16. When you finish, write the factorised form and say it is the square of an integer. Without that sentence you may lose the final mark even when the algebra is right.
Method: 1. Let the smaller integer be n and write the larger one using the difference. 2. Multiply them and add the extra number given in the question. 3. Factorise the quadratic, looking for a perfect square. 4. Write a conclusion saying it is the square of an integer.
A common proof asks you to show that a recurring decimal equals a fraction. Let x be the decimal and multiply by a power of 10 so the repeating blocks line up. For x=0.dot4dot5 the block has two digits, so multiply by 100 to get 100x=45.4545ldots. Subtracting the first equation removes the repeating tail.
The subtraction leaves 99x=45, so x=45/99, which simplifies to 5/11. The two equations must both be written out, and the subtraction must be shown. Many marks are lost by writing only the final fraction. A one-digit repeat needs a multiplier of 10, a two-digit repeat needs 100, and a three-digit repeat needs 1000.
Method: 1. Write x equal to the recurring decimal with several repeats. 2. Multiply by 10, 100 or 1000 so the repeating parts match. 3. Subtract the two equations to remove the repeating tail. 4. Solve for x and simplify the fraction.
The command word changes what the question wants. Prove means a general argument in algebra that covers every integer. Show that means the same, with a target to reach. Give a counter-example means one number that breaks the claim. A single counter-example is enough to disprove a statement, but no number of examples proves one.
For a claim such as the expression n^2+n is always even, factorise it as n(n+1). The two factors are consecutive, so one of them is even, and the product is even. Another way is to treat the cases n=2k and n=2k+1 separately. Either route earns marks if each step is stated clearly.
Method: 1. Read the command word: prove, show that, or give a counter-example. 2. To disprove, try small numbers until one fails and quote it. 3. To prove, write the general expression and factorise or expand it. 4. Finish with a sentence that matches what the question claimed.
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