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Back in the days when I was studying from John Thompson's First Grade Piano book, I had come across a peculiarity which I could not understand: in pages 13-15 there are pieces which are in the key of G Major. The instructions on page 13 ask to pay attention to the F notes which should be sharped. However, in none of the following 3 pieces an F note appears.

This has caused much confusion to me while studying, since I was looking for the F notes but couldn't find them.

Is there any reason for giving a key signature in a piece ?

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It's not particularly surprising that none of the pieces contained an F♯. First grade pieces tend to focus on the tonic, mediant, and dominant (with the supertonic and subdominant thrown in for runs and the like). The leading note is less likely to appear because beginning pianists are struggling with rhythm, identifying the limited notes they do know, etc. and having them learn to move their hands around the keyboard while still maintaining everything else is more difficult.

It is sad, though, that the instructions don't match the music.

But back to your question: Yes, there are many reasons for giving a key signature.

In the case of these songs, the purpose is limited. All it really lets you know is that the tonic is G (and that it's not G minor, etc.). But generally it provides a quick shorthand for the reader/player. What the G key signature really says is "F notes are actually F sharps". So when you see an F, you don't have to think about it; you're already in the "mode" of playing F sharps and automatically do it (this comes with practice, of course). On the other hand, putting a sharp on every F instead of in the key signature would be more mentally challenging; you'd have to recognize the note and then modify it. For this reason, accidentals are the exception rather than the rule. Key signatures make things simple.

We could, instead of referring to notes as sharps and flats, have them be their own notes. Instead of 7 notes A-G, we could have 12 notes A-L. That would eliminate accidentals and key signatures altogether. But again, it would be more complicated. Staffs would need to be larger, and recognizing notes based on their position (in a space / on a line) would be more difficult since there would be more spaces and lines to keep track of. This would also be more complicated if you weren't using equal temperament. When using just intonation an A♯ is different than a B♭, so you'd have more than 12 notes to represent (21, I believe).

The current method is a good balance between the two extremes of having a unique positional representation of every single note (like in the preceding paragraph) and having something like the following notation where you don't take advantage of vertical position at all:

Treble: A5--B5--C#5-----|--etc.

_ Bass: A3--------------|--etc.

In sum: Key signatures provide a quick shorthand so you can go into an automatic "Playing in G Major" mode where you play F as F♯ without needing to think about it. You could figure out what "mode" to play in without a key signature, but that would require deeper analysis of the piece and is frankly a waste of time when the music's composer already knows how they wrote it and can just put it down! It also allows for more efficient, readable notation than many alternatives.

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