Arrows fired from silvered bows DON'T count as silvered
The description in the PHB on silvered weapons states (bold added):
You can silver a single weapon or ten pieces of ammunition for 100 gp. (PHB, p. 148)
The option to silver ammunition does not make much sense if you could silver a bow for the same price, and use it to make every arrow you fire overcome resistance to attacks from weapons which are not silvered. The increase in flexibility (of which weapon you used the ammunition for) would not outweigh the potential cost. Especially since, according to the rules as written, you tend to recover only half of the ammunition that you use. (See "Ammunition", p. 146).
More importantly, there is nothing in the rules to suggest that arrows fired from a silvered bow count as silvered for the purpose of overcoming resistance or immunity. Contrast this with the rules on magical weapons (post Errata) which state:
If a magic weapon has the ammunition property, ammunition fired from it is considered magical for the purpose of overcoming resistance and immunity to nonmagical attacks and damage. (DMG, p. 140).
In the absence of similar text for silvered weapons, we have no reason to believe that arrows fired from a silvered bow count as a silvered weapon.
NOTE: As David Coffron pointed out in a comment, silvered weapons are not automatically "considered magical." There are some creatures that have resistance or immunity to "bludgeoning, piercing, and slashing from nonmagical attacks that aren't silvered." Attacks with silvered weapons or magical weapons will overcome this specific resistance or immunity. But this does not mean that magical weapons count as silvered, or that silvered weapons count as magical.