Mebbid
Aquarium Advice Addict
I wouldn't feed your Betta only once a week. Your tank is small enough you can do a water change easily and with Betta Water in case of a ammonia spike. Don't use tap, go to wal-mart or pet smart they sell Betta water very cheaply. I did use this water for my Betta (the one that jumped) and had no issues with cycling. Mostly because I would change about 10% each week. For a three gallon that would be .3 gallons, lol.
CaribSea Ready-Water - Fish Care - Fish - PetSmart
One of them lasted me for two month. I never used tap water for my Betta tank, and I never used tap for my saltwater. A cheaper method is just go to your supermarket and buy the filtered R/O water they sale in the machines. 3 gallons of that shouldn't cost you much, hardly over a dollar after you buy the jugs.
Trust me, using reverse osmosis water will save you so much trouble in the long run!
Your guy is still a growing Betta, you want to feed him well so he can grow nice and big with great color. Not feeding him at this point does nothing and does nothing to help your tank cycle. Just watch the water conditions and if the ammonia goes up, do a 90% water change. You won't harm your cycle process by doing a big water change once a week. The Bactria is in the rock and decorations, not in the water.
Unless your tap water is awful then this water isn't really necessary at all. It's basically the same thing as tap water with prime or stress coat added in.