GCSEFurther MathsAQAFlashcards2. Algebra & FunctionsAlgebraic Proof

Algebraic Proof (AQA GCSE Further Maths): Flashcards

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FrontAlgebraic Proof
Complete the algebraic forms, where $n$ is any integer:
An even number: $______$
An odd number: $______$
The integer straight after $n$ : $______$

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Complete the algebraic forms, where $n$ is any integer:
An even number: $______$
An odd number: $______$
The integer straight after $n$ : $______$
The completed forms are:
An even number: $2 n$
An odd number: $2 n + 1$
The integer straight after $n$ : $n + 1$
Use as few letters as possible, but a different letter such as $2 m$ for an unrelated even number.
How do you prove that an expression is always even?
Show that it can be written as $2 \times (\dots)$ , with an integer inside the bracket.
The same idea proves a multiple of any $k$ : write the expression as $k \times (\dots)$ .
True or False?
$2 n + 3$ represents an odd number for every integer $n$ .
True.
$2 n$ is even, and adding an odd number to an even number always gives an odd number.
$2 n + 1$ is the usual form, but $2 n + 3$ and $2 n - 1$ are just as valid.
Why is testing a few numbers not a proof?
Because a proof has to hold in every case, and examples only show it works for the ones you tried.
Using a letter such as $2 n$ stands for every even number at once, which is what makes the argument general.
Prove that the difference between the squares of two consecutive even numbers is divisible by 4.
Take the numbers as $2 n$ and $2 n + 2$ , so the difference of their squares is ${(2 n + 2)}^{2} - {(2 n)}^{2}$ .
Expanding gives $8 n + 4 = 4 (2 n + 1)$ , which is 4 times an integer and so divisible by 4.
If $p$ is prime, what are the only ways to write it as a product of two positive integers?
Only $1 \times p$ and $p \times 1$ , because a prime's only factors are 1 and itself.
That is often the step that finishes a proof about primes.
Why does writing $x^{2} - 6 x + 11$ as ${(x - 3)}^{2} + 2$ prove it is always positive?
Because a squared bracket is never negative, so ${(x - 3)}^{2} \geq 0$ and the whole expression is at least 2.
Since 2 is itself positive, the expression is positive for every value of $x$ .

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