In a way that shows you are unwilling or resentful about giving something or doing something.
From 'begrudge' (be- + grudge), which comes from Middle English 'grudgen.' The suffix '-ingly' converts the verb into an adverb form. The prefix 'be-' intensifies the base word 'grudge,' which means to resent or be unwilling.
The prefix 'be-' appears in tons of English words (belittle, besmirch, beset) and originally meant 'around' or 'about'—but over time it became an intensifier that basically adds attitude to a word. Begrudgingly is like saying you're surrounded by grudge!
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