Edexcel IGCSE Math A 2.2.HE Algebraic Proof

Practise representing consecutive or restricted integers algebraically and rearranging results to prove divisibility or parity.

Exam points

represent consecutive integers, even numbers or multiples using n-based expressions
expand and simplify both sides until the claimed difference or equality is visible
factor the result to exhibit an integer multiple or the form 2k+1 for oddness

[Maximum number: 3]

Here are the first four terms of a sequence of fractions.

\frac{1}{1} \quad \frac{2}{3} \quad \frac{3}{5} \quad \frac{4}{7}

The numerators of the fractions form the sequence of whole numbers $1234 \ldots$ The denominators of the fractions form the sequence of odd numbers $1 \quad 3 \quad 5 \quad 7 \ldots$

Using algebra, prove that when the square of any odd number is divided by 4 the remainder is 1

Let an odd number be 2n-1.
$(2n-1)^2=4n^2-4n+1$ $=4(n^2-n)+1$
Therefore when the square is divided by 4, the remainder is 1.

M1 for using $(2n-1)^2$ or $(2n+1)^2$ .
M1 for writing the expression as a multiple of 4 plus 1.
A1 for a correct conclusion.

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2.2.HE Algebra to support and construct proofs question 1