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Welcome to GCSE Edexcel Maths revision.

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Topic M 17: Algebraic arguments and proof.

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This video covers Higher tier.

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It includes the shared content and the labelled Higher extensions.

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An equation is true for particular values: 2 x plus 1 equals 7 holds when x equals 3.

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An identity holds for every allowed value, for example 2 multiplied by open bracket x plus 3 close bracket is identically equal to 2 x plus 6.

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To show expressions are equivalent, expand or factorise and collect terms until they have the same form.

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Testing a single value cannot prove an identity.

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Worked example: 3 multiplied by open bracket x plus 2 close bracket minus x equals 3 x plus 6 minus x equals 2 x plus 6 equals 2 multiplied by open bracket x plus 3 close bracket ,

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so the two expressions are equivalent for every x.

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A counterexample disproves a universal claim.

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The statement all prime numbers are odd is false because 2 is prime and even.

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Use precise mathematical reasons in an argument.

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A diagram that looks correct or a few numerical examples are evidence to investigate, not a proof for every case.

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Consecutive integers can be written n, n plus 1, n plus 2; an even integer as 2 n; an odd integer as 2 n plus 1, with n an integer.

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To prove a divisibility statement, express the result as the divisor times an integer.

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Define the integer variables and show each step clearly.

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Worked example: The sum of two odd integers is open bracket 2 m plus 1 close bracket plus open bracket 2 n plus 1 close bracket equals 2 multiplied by open bracket m plus n plus 1 close bracket .

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Since m plus n plus 1 is an integer, the sum is even.

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Worked example: The difference of consecutive squares is open bracket n plus 1 close bracket squared minus n squared equals 2 n plus 1, which is odd for every integer n.

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Worked example: Three consecutive integers sum to n plus open bracket n plus 1 close bracket plus open bracket n plus 2 close bracket equals 3 multiplied by open bracket n plus 1 close bracket ,

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so their sum is a multiple of 3.

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For an odd square, open bracket 2 n plus 1 close bracket squared equals 4 n squared plus 4 n plus 1 equals 4 n multiplied by open bracket n plus 1 close bracket plus 1.

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Hence its remainder on division by 4 is 1.

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Distinguish a proof from solving an equation.

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Proving an identity must not impose a special value of x; every operation must be valid across the stated domain.

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To disprove x squared is always greater than x, use x equals 1 over 2 to x squared equals 1 over 4 is less than 1 over 2 .

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Consider values beyond positive integers when a claim concerns real numbers.

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That completes Algebraic arguments and proof.

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Revisit the notes and test yourself on the revision website.