Is there a word to describe the ability to be able to complete tasks without necessarily needing instructions, i.e. finding it out yourself.
It's for a personal profile in a CV.
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I often see self-motivated or self-directed used on CVs I receive.
self-starter Informal. a person who begins work or undertakes a project on his or her own initiative, without needing to be told or encouraged to do so.
Although it say "informal" there, I see no reason not to use it in your CV.
I don't think that a single word incorporates both the "not needing directions" aspect of your description (which FumbleFingers addresses with self-starter) and the "solving an issue" aspect. If you want to cover both, you might try a phrase such as "creative, independent problem-solver."