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Algebraic proof
There are some key words that you need to remember the definition of:
Integer
a whole number
Sum
add
Product
multiply
Difference
subtract
Algebraic proof uses specific notation:
n
A number
2n
An even number
2n + 1
An odd number
n, n+1, n+2 etc.
Consecutive numbers
2n, 2n+2, 2n+4 etc.
Consecutive even number
2n+1, 2n+3, 2n+5 etc.
Consecutive odd number
Example. Prove that the sum of the squares of any two consecutive odd
numbers is even.
First I underline all the key information.
We write two consecutive odd numbers as 2n + 1 and 2n + 3
The question wants the squares of these:
(2n + 1)2 = (2n + 1)(2n + 1) = 4n2 + 4n + 1
(2n + 3)2 = (2n + 3)(2n + 3) = 4n2 + 12n + 9
Remember that (2n + 1)2
means double brackets
We want the sum of the squares:
4n2 + 4n + 1 + 4n2 + 12n + 9 = 8n2 + 16n + 10
Finally, we are trying to prove that this answer is even. To show something is
even, each term must be divisible by 2. The easiest way of showing this is to
factorise:
8n2 + 16n + 10 = 2(4n2 + 8n + 5) each term has a factor of 2, so the sum of the
squares of any two consecutive odd number is even.
Be careful when subtracting a bracket e.g. (n + 1)² - (2n + 3). We are subtracting
each term in the bracket, so (n + 1)² - 2n - 3
1. 8 Prove that (3n + 1)² - (3n - 1)² is a multiply of 6 for all positive integer
values of n.
2. 10 Prove that (5n + 1)² - (5n - 1)² is a multiply of 5 for all positive integer
values of n.
3. 13 Prove that (2n + 1)² - (2n - 1)² -10 is not a multiply of 8 for all positive
integer values of n.
4. 15 Prove that (n + 1)² - (n - 1)² + 1 is always odd for all positive integer
values of n.
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