ましょうかShall we, Shall I

Both ～ましょうか and ましょう are polite expressions used to suggest or propose an action. ましょう is more assertive, similar to 'Let's (do this)' in English. In contrast, ～ましょうか is softer and seeks the listener's approval or agreement, like 'Shall I...?' or 'Shall we...?' ましょう can also function as a gentle command, such as when a teacher says 'Let's be quiet' to mean 'Be quiet.'

ましょう and ませんか are both polite forms based on the ます form and used to make suggestions or invitations. ましょう typically suggests doing something together (e.g. 食べましょう = 'Let's eat'), while ませんか invites the listener to join (e.g. 食べませんか = 'Won't you eat [some]?') ましょう is more assertive, while ませんか is softer and more inviting.

ましょうか and Verb [よう] are both volitional forms, with ましょうか being polite and Verb [よう] (sometimes with か) casual. ～ましょうか is used to make polite suggestions like 'Shall we...?' (e.g. 行きましょうか = 'Shall we go?') 行こうか is more casual and introspective, often expressing the speaker's internal thoughts (e.g. 'Should I go?')