THe order of the sharps is: F C G D A E B so when you go to the right of the circle of fifths that I've transcribed before you have that
G major has 1 sharp and as the first one is F the G major scale would have F# leaving: G A B C D E F# G
D major has 2 sharps and as G major had F# you'll add one more sharp to this scale and that would be C# and the result would be: D E F# G A B C# D
If you keep adding sharps to the next keys (A, E, B, F# and C#) you'll find that the next key has the previous sharp and one more (using the order that I've written at the first part of the post)
When you talk about flats is the same thing but backwards, the order that the flats get into the keys is the inverse of the sharp order, and that would be: B E A D G C F
So as F major is the first key to the left of C major in the circle of fifths that key has 1 flat and that would be Bb leaving the scale: F G A Bb C D E F
The following scale in the circle would be Bb and that has the previous alteration (Bb) and the next one Eb, so Bb major would be: Bb C D Eb F G A Bb
And as in the example with the sharps the list of alterations moves along with the keys.
This circle of fifths is made to memorize all the keys but in a easier way, because if you do the math to the major scale like TTSTTTS in every key the result is the circle of fifths. The theory behind this nmemotechnique rule is really easy.
You go from C to the right adding up fifths and you add one sharp by one key and as you do that you keep adding the sharps of the previous keys.
As you go to the left of the circle you go down fifths and you add flats one per key and the next key add one plus the previous that had the previous key.
I hope this clarify the way to read the circle
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