I bought a 5ft areca palm about a month ago. About two weeks after I purchased it, I noticed some leaves had spots and others were turning brown. Initially I thought it was normal but more leaves seem to be showing these symptoms. The palm is indoors, in 70 degree temperature, and sitting about 3 meters from the living room windows (windows face back patio and let in medium sunlight, though doesn't hit palm directly). I water it every other day (about a cup of water) and don't use fertilizer, feed or anything else.
First thing to do is correct your watering regime - little and often is not a good idea, it's better to let the plant dry out slightly between waterings, and water thoroughly when you do water. If the soil in the pot feels dry to the touch, give it a litre or two of water. If there's any water left in the outer pot after 30 minutes, empty that out so that the palm is not sitting in water. Then repeat this amount of water when the soil on the top of the pot feels dry to the touch, but not so dry it's shrunk from the sides of the pot - probably about once a week might be about right in that sort of temperature, but always check the soil on the top first. If you have a water softener in your home, then don't use tapwater, the sodium left behind in softened water may cause yellowing of the leaves.
I don't know where you are in the world -you say you bought your palm a month ago, but if your weather was cold, a bit of transition shock from being purchased, brought home and being moved between cold and hot might have caused some damage to occur.
Cut out any fronds that are completely brown at the base, and check the whole plant over for anything that shouldn't be there, like a spider mite infestation. More information on how to care for these palms as houseplants can be found in this article: Indoor Areca Palm: yellow Leaves.
@Bamboo is as usual completely right. What I would like to add is repotting might be the solution. I would:
- get a larger pot
- pour soil in
- make an indentation the size of the current pot
- water the current pot as to keep the soil together
- drop the rootball into the larger pot.
Adhere to the watering advice in the above answer. Also - given my hypothesis is true - pruning dead tissue is purely aesthetical and at best unrelated to the plant's health.