Awww, bink, that sounds lovely! I might take a look.
Reallytired, now I am working my way through the algebra chapter, the only bit I actually can't move forward on is on using algebra to prove general statements. Everything else seems to have clicked as I have remembered my GCSE.
This isn't a question; it's an example, but I fail to see how they arrived at the answer:
In maths it is possible to use summing to prove general statements and also conjectures. A conjecture is a hypothesis, something that has been surmised or deduced.
One such instance is stating that adding consecutive odd numbers, starting at 1, will result in the square numbers, i.e.:
1 = 1 = 1squared
1+3 = 4 = 2squared
1+3+5 = 9 = 3squared
1+3+5+7 = 16 = 4squared (I am following well so far! Easy so far!)
In order to prove this, it is necessary first to construct the general sequence of odd numbers (use the difference methos above if necessary):
position: 1 2 3 ... (n-2) (n-1) n
term: 1 3 5 ... (2n-5) (2n-3) (2n - 1)
If the sum of the first n odd numbers is S, then:
S = 1+3+5...+(2N-5) + (2N-3) + 2N-1)
The expression that is equal to S can be written in reverse order without changing the value due to the commutative nature of addition:
S + (2n-1) + (2n-3) + 2n-5) + ... + 5 + 3 + 1
If these two values for S are added together a value for 2S is achieved:
2S=(1+(2n-1) + (3+(2n-3) + (5+(2n-5) + ... +((2n-5)+5) + ((2n-3)+3) + ((2n-1)+1)
2S= 2n+2n+2n+...+2n+2n+2n
2S=nx2n
2S=2nsquared
S=nsquared
This shows that the sum of consecutive odd numbers starting at 1, will result in a square number being generated.
OK, I've done differences, quadratic expression, general statements, substitution and simplification. This one page is peeing me off!
I have done my Science and English audits and got 90%+ on each section of those. All the other parts of my maths are fine.