hello I have been playing about 6 months...I know my basic music theory, major scales, penatonics, circle of fifths, chord construction etc.

I am starting to learn arpeggios...im just checking im doing it right. so im trying to build an A major arpeggio using the notes A C# E (1st 3rd 5th). here is the tab.


all of the notes are A, C#, E so this is right?

the other problem I got is...I know how to do chord construction, how to make into minor, augmented, suspended, dimished etc but I can't really apply it to my playing very well because it takes me a 5-10 seconds to work out to put which fingers where.

is there any ways to increase how fast I can learn the formulas and learn better the note positions so it's just like instinctive, like I could just play any chord and not think about it at all?

I suppose it's just like alot of stuff on guitar, just got to practice this skill?
The arpeggio is correct, and yes, it's just practice.

A lot of people learn the common shapes of barre chords and make them in reference to the root notes on the fretboard. I'm assuming this is what you're doing, and if so, just practice more. If you're following some other sort of system, consider learning the common shapes to help guide you. It's of course best to understand why those shapes are correct, and it seems you're on your way there.
And no, Guitar Hero will not help. Even on expert. Really.
just a tip on the arpeggio, ending on the 3rd is a little "unstable". this can work in your favor at times if that's your goal, but typically you'll want to end on the root unless you're transitioning into another arpeggio directly after.
modes are a social construct
About your second problem:
When playing a chord, you need to know the notes you are playing, relatively to the root. By that I mean that when playing a Em (022000), that the lowest string (E) is your root note (1) and so on. For the second string: A-5, third: D-3, G-1, B-5, e-1. If you know what I mean. If you look at this you see that only the D-string has the third in it. So if you want to change the chord into E, you'll need to raise that note with a half tone: 022100.

So yeah, it basically takes practice to be able to know/recognize which string is what note.
you should practice it a lot more. what i'd do is: practice strumming the chord, trying to break the notes up in your head (like try to hear the pitches, or even sing them out loud) and then play the arpeggio while singing the note names (or solfege syllables if you use them), then play the chord again. then repeat it for a chord a fourth up, until you go through all twelve keys--then do it a lot more.
so i'd go like
starting on a C chord
strum the chord. sing the notes C-E-G-E-C, then play the two octave arpeggio, and then go to F and repeat it, etc.
also, sitting down and writing stuff out in all 12 keys can hurt.
all the best.
(insert self-aggrandizing quote here)