The plain volitional form expresses intent or invitation — the casual equivalent of ましょう. Used in patterns like 〜と思う (I think I'll...) and 〜とする (be about to / try to).
u-verb: -u → -ō | ru-verb: drop る + よう | する → しよう, くる → こよう
今夜は早く寝よう。
Kon'ya wa hayaku neyō.
I'll go to bed early tonight. / Let's sleep early.
寝る → 寝よう (ru-verb: drop る + よう)
そろそろ行こうか。
Sorosoro ikō ka.
Shall we get going?
行く → 行こう (u-verb: く→こ + う) + か
今年こそ日本語を頑張ろうと思っています。
Kotoshi koso nihongo o ganbarō to omotte imasu.
I'm thinking of working hard at Japanese this year.
頑張ろう (volitional) + と思っています = I think I'll...
• Volitional uses: (1) casual suggestion/invitation (行こう!= let's go!); • Detail: (2) 〜と思う (I intend to / think I'll...); • (3) 〜とする (was about to / tried to: 寝ようとした = tried to sleep). • Pattern table: 書く→書こう, 飲む→飲もう, 食べる→食べよう, する→しよう, くる→こよう.
The JLPT rarely asks whether you recognise a pattern — it asks which of several similar patterns fits. These are the ones 意向形 is tested against.
ましょう is the polite form of the same volitional meaning (行きましょう). The plain volitional (行こう) is its casual counterpart used with friends or in self-talk.
〜たい states a personal DESIRE (食べたい = I want to eat). The volitional expresses DECISION/intent or an invitation (食べよう = let's eat / I'll eat) — a resolve to act rather than a wish.
つもり describes a firm, pre-formed plan (行くつもり). 〜(よ)うと思う uses the volitional for a decision often made just now or a softer intention.
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