Algebraic Proof (AQA GCSE Maths)

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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.
$2 n + 1$ , for example, where $n$ is an integer, is an expression for an odd integer.
Another possibility is $2 n - 1$ .
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 $n + 1$ , for example, where $n$ is an integer, are expressions for two numbers that are consecutive.
Another possibility is $n - 1$ 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 $2 n + 2$ , 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:
$n + (n + 1)$
This shows an integer, $n$ , being summed with the next (consecutive) integer, $(n + 1)$ .
True or False?
If $n$ is an integer, then $2 (n^{2} + n) + 1$ is odd.
True.
If $n$ is an integer, then $2 (n^{2} + n) + 1$ is odd.
It has the form of an odd number, $2 \times (integer) + 1$ , as $n^{2} + n$ is an integer.
True or False?
$2 (n + \frac{1}{2})$ represents an even integer, as it is written in the form $2 \times (. . .)$ .
False.
It is true that even numbers are written in the form $2 \times (. . .)$ but the part inside the brackets must be an integer.
The problem with $2 (n + \frac{1}{2})$ is that the part inside the brackets has a $\frac{1}{2}$ in it, so would never be an integer.
Alternatively, you could expand the brackets to see that $2 (n + \frac{1}{2}) = 2 n + 1$ 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 $3 n + 3$ are two consecutive multiples of 3.
${(3 n)}^{2}$ and ${(3 n + 3)}^{2}$ are squares of two consecutive multiples of 3.
${(3 n + 3)}^{2} - {(3 n)}^{2}$ is the difference between the squares of two consecutive multiples of 3.
So the answer is ${(3 n + 3)}^{2} - {(3 n)}^{2}$ .
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 $8 n + 5$ is odd (assuming $n$ is an integer)?
Odd numbers have the form $2 \times (integer) + 1$ .
One way show that $8 n + 5$ is odd is to rewrite $8 n + 5$ as $8 n + 4 + 1$ then factorise out a 2 to get $2 (4 n + 2) + 1$ .
This now has the form of an odd number as given above.
If you factorised 4 out instead to get $4 (2 n + 1) + 1$ , you would need an extra line of working to explain how to get to the form $2 \times (integer) + 1$ .
True or False?
If $p$ is a prime number and $p = a b$ where $a$ and $b$ are positive integers, then $a = 1$ and $b = p$ .
False.
This is almost true, but it is missing a second possibility.
If $p$ is a prime number and $p = a b$ where $a$ and $b$ are positive integers, then $a = 1$ and $b = p$ is one possible answer.
The other possible answer is $a = p$ and $b = 1$ .

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