As covered in our ましょう
is one of the conjugations of the auxiliary verb ます, and is a common (polite) way to suggest that something be done ‘together’.
Adding the particle か
to this expression simply changes the nuance from ‘let’s’, to ‘shall we?’, and is more of a direct question than ましょう
, this expression can be added to the ます stem form of any verb.
One thing to note with these examples is that they are ‘suggestions’, not ‘invitations’. In Japanese there is a slight difference between the two, in the same way that there is in English. If you want to invite someone to do something, the most natural way would be to use ませんか
, a phrase that is similar to ましょうか
The difference here is that ‘won’t you do (A)?’ has a stronger nuance of being a direct invitation, while ましょうか
is closer to just ‘let’s do (A)!’