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Algebraic Proof (Edexcel IGCSE Maths A (Modular))

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  • FrontAlgebraic Proof

    Write down an expression for an even integer.

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  • Write down an expression for an even integer.

    2 n, for example, where n is an integer, is an expression for an even integer.

    Other letters could also be used.

  • Write down an expression for an odd integer.

    2 n plus 1, for example, where n is an integer, is an expression for an odd integer.

    Another possibility is 2 n minus 1.

  • Write down an expression for an integer that is a multiple of 7.

    7 n, for example, where n is an integer, is an expression for an integer that is a multiple of 7.

  • Write down expressions for two numbers that are consecutive.

    n and n plus 1, for example, where n is an integer, are expressions for two numbers that are consecutive.

    Another possibility is n minus 1 and n.

  • Write down expressions for any two even numbers.

    2 n and 2 m, for example, where n and m are different integers, are expressions for any two even numbers.

    It is not 2 n and 2 n plus 2, as this would mean two consecutive even numbers.

  • What is the first line of algebra when proving that the sum of two consecutive integers is odd?

    The first line of algebra when proving that the sum of two consecutive integers is odd is:

    n plus open parentheses n plus 1 close parentheses

    This shows an integer, n, being summed with the next (consecutive) integer, open parentheses n plus 1 close parentheses.

  • True or False?

    If n is an integer, then 2 open parentheses n squared plus n close parentheses plus 1 is odd.

    True.

    If n is an integer, then 2 open parentheses n squared plus n close parentheses plus 1 is odd.

    It has the form of an odd number, Error converting from MathML to accessible text., as n squared plus n is an integer.

  • True or False?

    2 open parentheses n plus 1 half close parentheses represents an even integer, as it is written in the form 2 cross times open parentheses... close parentheses.

    False.

    It is true that even numbers are written in the form 2 cross times open parentheses... close parentheses but the part inside the brackets must be an integer.

    The problem with 2 open parentheses n plus 1 half close parentheses is that the part inside the brackets has a 1 half in it, so would never be an integer.

    Alternatively, you could expand the brackets to see that 2 open parentheses n plus 1 half close parentheses equals 2 n plus 1 which is odd (assuming n is an integer).

  • How do you write an algebraic expression for the difference between the squares of two consecutive multiples of 3?

    3 n and 3 n plus 3 are two consecutive multiples of 3.

    open parentheses 3 n close parentheses squared and open parentheses 3 n plus 3 close parentheses squared are squares of two consecutive multiples of 3.

    open parentheses 3 n plus 3 close parentheses squared minus open parentheses 3 n close parentheses squared is the difference between the squares of two consecutive multiples of 3.

    So the answer is open parentheses 3 n plus 3 close parentheses squared minus open parentheses 3 n close parentheses squared.

    Note also how you usually write the bigger number subtract the smaller number, so that the overall result is positive.

  • How could you show that 8 n plus 5 is odd (assuming n is an integer)?

    Odd numbers have the form Error converting from MathML to accessible text..

    One way show that 8 n plus 5 is odd is to rewrite 8 n plus 5 as 8 n plus 4 plus 1 then factorise out a 2 to get 2 open parentheses 4 n plus 2 close parentheses plus 1.

    This now has the form of an odd number as given above.

    If you factorised 4 out instead to get 4 open parentheses 2 n plus 1 close parentheses plus 1, you would need an extra line of working to explain how to get to the form Error converting from MathML to accessible text..

  • True or False?

    If p is a prime number and p equals a b where a and b are positive integers, then a equals 1 and b equals p.

    False.

    This is almost true, but it is missing a second possibility.

    If p is a prime number and p equals a b where a and b are positive integers, then a equals 1 and b equals p is one possible answer.

    The other possible answer is a equals p and b equals 1.

Previous:Rearranging FormulasNext:Solving Inequalities

    1. Topic 1

      Prime Factors, HCF & LCM

    2. Topic 2

      Standard Form

    3. Topic 3

      Using Percentages

    4. Topic 4

      Compound Interest & Depreciation

    5. Topic 5

      Ratio Toolkit

    6. Topic 6

      Ratio Problem Solving

    7. Topic 7

      Direct & Inverse Proportion

    1. Topic 1

      Rearranging Formulas

    2. Topic 2

      Algebraic Proof

    3. Topic 3

      Solving Inequalities

    4. Topic 4

      Simultaneous Equations

    5. Topic 5

      Forming & Solving Equations

    6. Topic 6

      Sequences

    1. Topic 1

      Functions

    2. Topic 2

      Graphing Inequalities

    3. Topic 3

      Transformations of Graphs

    4. Topic 4

      Differentiation & Gradients

    1. Topic 1

      Angles in Polygons & Parallel Lines

    2. Topic 2

      Bearings, Scale Drawing & Constructions

    3. Topic 3

      Circle Theorems

    4. Topic 4

      Volume & Surface Area

    5. Topic 5

      Similarity & Geometrical Proof

    6. Topic 6

      Area & Volume of Similar Shapes

    1. Topic 1

      Vectors

    2. Topic 2

      Transformations

    1. Topic 1

      Statistics Toolkit

    2. Topic 2

      Cumulative Frequency Diagrams