Just started with this build and its being improved when I have time with it

I start with a Long Sword and a Health Potion. The Long Sword for the passive Twin Disciplines
On my first recall i get my self Boots and if i can afford it, Amplifying Tome.
After that we upgrade our Boots to Mercury's Treads, because the Tenacity it gives is just lovely.
Next up, u get to choose. Either go with Phage which gives Health, Attack damage (more Attack damage = More Spell vamp) and a nice slow. Or u go with Hextech Revolver which gives 40 Ability power and 15% Spell vamp. I usually go with a Phage for the Health. Next up is Trinity Force. Yes, the biggest, most expensive, Hardcore item League of Legends have. but!. Where to start? well, since we got a Phage we only need a Sheen and a Zeal. I'd says start with a Sheen and then a Zeal. Because NOTHING is better then that 100% extra damage u get from Sheen's passive. After that i the enemy team will probable start focus you, so we need some Health. But we cant JUST get Health, we need to do some Damage to. So the best item for us is..... Rylai's Crystal Scepter which is just great, because its gives 80 AP and 500 health PLUS it makes so ALL of your ability's slow! isn't that just great. After that we build our Hextech Revolver in to a Hextech Gunblade. 70 AP, 40 AD, 15% Lifesteal and 20% Spell vamp. If u get this far, a Rabadon's Deathcap is next. 140 AP AND increase your Ability power with 30%! And since we got Archmage we get 35% extra AP. Look in the Radadon's Deathcap section for how much it will increase.
Rarely we wont get this far, but now its time for Void Staff! Cool name, don't you think? The name isn't the only think that is cool about this item, it gives 70AP AND 40% Magic Penetration! Thats just AWESOME!

In this section i will go over mostly math

Rabadon's Deatchcap

Items up to Rabadon's Deathcap ( Trinity Force, Rylai's Crystal Scepter and Hextech Gunblade) gives you 180AP
30+80+70=180AP.
lets say your level 18, then u get 27.54+14,85=29,7 AP from Runes.
Then 3+18+5=26 AP from masteries. ( Archmage not count with)
180+29,7+26=235.7 AP Before Rabadon's Deathcap

235,7+140=375,7 AP 375.7 ~ 376 AP without Rabadon's Deathcap's Passive and Archmage

375.7*0,65=507,195 AP (With Rabadon's Deathcap's Passive and Archmage
As you can see 35% is ALOT of bonus AP. we went from 236AP and end up with 507AP
Thats around 51% more AP from ONE item!

Magic Penetration

Now Magic penetration is just great, just really awesome. You can look at most builds and they say the same. Penetration is the best. But why?

Well, magic penetration means u penetrate someones magic resistance ( You ignore there magic resistance).
But why is that good? well, if you go around and ''one-hit'' everyone they will not just sit there and take it. No, they will do something about it. and if they cant kill you, they will protect them selfs instead by getting magic resistance. now, magic resistance is hated by ALL AP champs, since it Resists there Magic damage. So if someone gets 400 magic resistance, it will be the same as before, but this time. We cant Defend against it, and what have we left then? we cant kill him and there is nothing to protect us from. And there is where Magic Penetration comes in handy. Magic penetration ignores there magic Resistance and we can do damage to him again. Great, right? there is just one problem... if we get only magic penetration we will not get much AP. and without AP we don't much damage to them after all.
So where is the right balance? between high Magic penetration and high AP. Riot have a solution to this problem. Its Void Staff and Arcane Knowledge !

Time for Math! (yay -.-)

NOTE: Magres = Magic Resistance. Magicpen = Magic Penetration

Lets say we face a Galio and he has 300 magic resistance. there isn't much we can do to him then. But with help from Arcane Knowledge and Void Staff its no problem.

So from Runes we get 8,55 magicpen.
so we will bring he's magres 300-8,55=291,45 magres. He still got way to much for us to do some real damage on him.
So we add our Mastery Arcane Knowledge
Now we got 8.55|10% magicpen.
300-8,55*0.9=262.305 magres. Still much

Now we add Void Staff which gives 40% magicpen
we now got 8,55|50% magicpen
300-8,55*0,5 now he got 145.725 magres. Now we hurt him, and we hurt him good.

lets say we also have 9x Greater Glyph of Magic Penetration and 3x Greater Quintessence of Magic Penetration
that will give us 5.67 from Quintessence and 5.13 from Glyphs and plus we our 8,55 from Marks

5,67+5,13+8,55=19,35 magicpen
now we add Void Staff and Arcane Knowledge which gives us:
19.35|50% magicpen
so now we meet Galio again
300-19,35*0,5=140.325 magic pen. Not so much less then if we only had 9x Greater Marks of Inisght