How is a + A ^ 2 + A ^ 3 +... + A ^ n = a * (a ^ n-1) / (A-1) derived

How is a + A ^ 2 + A ^ 3 +... + A ^ n = a * (a ^ n-1) / (A-1) derived

Let the left formula be x and the left multiplied by A,
Then subtract the formula on the left, that is, ax-x = many terms can be eliminated, see?
Then there are the first two terms left. Divide the two sides by a - 1, and you get

（a-b）（a^(n-1)-b^(n-1)）=(a-b)^2(a^(n-2)+a^(n-3)b+…… +How to deduce AB ^ (n-3) + B ^ (n-2))

What I don't understand most is what the ellipsis omits

1 / 2,1,5 / 4,7 / 5,3 / 2 for a ^ n

General term (2n-1) / (n + 1)