danken
to thank
To express one's gratitude to someone for something.
Ich möchte Ihnen herzlich für Ihre Unterstützung danken.
I would like to sincerely thank you for your support.

to thank
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to thank
To express one's gratitude to someone for something.
Ich möchte Ihnen herzlich für Ihre Unterstützung danken.
I would like to sincerely thank you for your support.

The verb 'danken' (to thank) requires a special sentence structure: the person you are thanking is always in the dative. The reason for the thanks is expressed with the preposition 'für' (for) and the accusative.
Ich danke dir für deine Hilfe.
I thank you for your help.
'Danken' (to thank) belongs to a word family with the noun 'der Dank' (thanks/gratitude) and the adjective 'dankbar' (grateful). All of these words express gratitude.
Vielen Dank für das tolle Geschenk!
Many thanks for the great gift!
To politely decline an offer, people often say 'Nein, danke' (No, thank you). Sometimes just 'Danke' (thanks) is enough, with the intonation and situation making it clear that it is a refusal.
Möchten Sie noch Kaffee? – Nein, danke, ich habe genug.
Would you like more coffee? – No, thank you, I've had enough.
The idiom 'jemandem etwas zu danken haben' (to have someone to thank for something) means that someone is responsible for a result. It is often used ironically for negative consequences.
Dass wir den Zug verpasst haben, haben wir nur dir zu danken.
We have only you to thank for the fact that we missed the train.
to accept / assume
'annehmen' and 'danken' can form an opposite pair when it comes to an offer. 'annehmen' means that you accept the offer or take something. Here, 'danken' does not just mean saying thank you, but politely declining, often in phrases like 'Nein, danke' (No, thank you) or 'Ich danke' (No, thank you). In everyday language, 'ablehnen' is clearer, but 'danken' sounds polite and somewhat brief. Context is important: as a normal verb, 'danken' usually means showing gratitude, not declining.

Ich danke, aber ich habe heute leider keine Zeit.
No, thank you, but unfortunately I don't have time today.
Maria nimmt die Einladung zum Abendessen gern an.
Maria is happy to accept the invitation to dinner.