Yes, 1-3-5 are chord tones of a plain old major chord. Check out the
CAGED lesson I did for more on that.
I think often people think that playing arpeggios means holding a chord shape and picking notes separately. That's not really it. It's using the notes from chord shapes as single notes, playing through them melodically as a series of single notes that can be scattered all over the fretboard. The more extended the chord, the more you have to choose from. Click on the "Power of Chord Tones" link in the moving ticker tape of my lessons above to watch a couple of examples of playing arpeggios that don't look at all like playing through chord shapes ... even though that's exactly what it is.
Here are links to those two "Power of Chord Tones" lessons:
The Power of Chord Tones
The Power of Chord Tones 2