Define algebraic proof , and say how it differs from testing values.

Algebraic proof means proving a result using algebra , so that it is shown to hold for every value at once. Testing individual values only shows the result works for the ones you happened to try, which is not a proof.

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Algebraic Proof (Edexcel IGCSE Maths B): Flashcards

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Define algebraic proof, and say how it differs from testing values.

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Define algebraic proof, and say how it differs from testing values.
Algebraic proof means proving a result using algebra, so that it is shown to hold for every value at once.
Testing individual values only shows the result works for the ones you happened to try, which is not a proof.
In an algebraic proof, what do letters such as $n$ and $k$ stand for?
They stand for any positive integer, rather than for any number at all.
That is what lets one piece of algebra prove a result for every whole number at the same time.
In these forms $k$ stands for a whole number. Fill in the two missing parts.
$even means 2 \times______$
$odd means 2 k \pm______$
The completed forms are:
$even means 2 \times k$
$odd means 2 k \pm 1$
Both are reached by rewriting and factorising the expression until it takes that shape, and $k$ has to be a whole number for the argument to work.
How do you prove that an expression is a multiple of 7?
Rewrite it until it can be factorised into the form $7 \times k$ , where $k$ is a whole number.
For example $7 n^{2} + 14 n = 7 (n^{2} + 2 n)$ , and $n^{2} + 2 n$ is a whole number whenever $n$ is.
How do you prove that an expression is a square number?
Rewrite it until it can be put in the form $k^{2}$ , where $k$ is a whole number, which usually means factorising it.
For example $n^{2} + 6 n + 9 = {(n + 3)}^{2}$ .
True or False?
$2 (n + \frac{1}{2})$ is even, because it is 2 multiplied by something.
False.
For $2 \times (something)$ to be even, that something has to be an integer, and $n + \frac{1}{2}$ is not.
In fact $2 (n + \frac{1}{2}) = 2 n + 1$ , which is odd.
A proof involves ${(6 n + 5)}^{2} - {(6 n - 5)}^{2}$ . Why is the difference of two squares quicker here than expanding both brackets?
It rewrites the expression as $((6 n + 5) + (6 n - 5)) ((6 n + 5) - (6 n - 5))$ , and each bracket collapses immediately to $12 n$ and $10$ .
That reaches $120 n$ without either square ever being expanded.
An expression simplifies to $\frac{2 n + 1}{2}$ . Why can it never be an integer?
The numerator $2 n + 1$ is odd, and an odd number divided by 2 is never a whole number.
Splitting the fraction shows the same thing: $\frac{2 n + 1}{2} = n + \frac{1}{2}$ , an integer plus a half.

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