Why these?

Quintences : U want to start with a good amount of ap..
Glyphs : It provides u with 27.54 ap at 18 lvl, u are an ap carry u must have a lot of ap
Seals : I think Gragas uses a lot of mana for his abilities, u need this mana regen.. U can also take hp/lvl or ap/lvl but i think that mana rgn. is better for gragas...
Marks : U need these!!

Drunken rage provides with a good amount of mana, makes u tanky and also increase the damage of body slam!!

But if u have no problem with mana and ur enemy dont do a lot of dmge u can maximize body slam 1st..

start item :
core items :
if u have a hard time on mid u can make this item otherwise if u doing great dont consider about taking this wonderfull item
and dont leave the base without taking these they will save u out of a lot of trouble.. Wards is a must to take by all the team, not only from support!!! U must take them until the game ends.. dont think that is a job for support to take a walk at all map and ward it!!

final items :
if u take 1st this consider about taking 1st this and then this cause abyssal + deathfire making a good combo, u will not believe how easily
u can kill the enemies especially ad + ap carries!!

WHY THESE ITEMS ?

If u a ap carry u must buy it! It provides u the most ap from all items..

U are ap carry u want magic penetration!!

U would telling me that the range of this item's range of his passive is ( < ) of ur abilities ( q + r ), but consider that u have body slam and if do that ta an enemy then u will hit him with the barrels on the range of ur item's passive... Also it provides u with a good amount of ap and magic resistance... I would prefer this items from others!

I love this item!! When i have it i am thinking that i have 2 ulties!! Consider that u have 500 ap!, that means that ur item will deal damage equal to 45% of the enemys health!! Is this brilliant or i am stupid???!!! U destroy ur enemy's half health with only an item!!! Also it gives u ap and cooldown resist, isnt it brilliant?

It would save u a lot of times!! consider that u are the ap carry, u are the enemy's target!! U must survive and deal dmge to the enemies!!
This item gives u 100 ap(amazing), 50 armor and it has this wonderful passive that u can be invulnerable and untargetable from ur enemies for 2 seconds!!
If ur enemy deals a lot of physical damage consider buy it after rabandons...

More magic penetration = more damage ... Its like u facing a naked enemy( no magic resist :P )