Derivation of the formula for the first n sums of the proportional Series

Set the first n of the proportional series to S (n), the first of the proportional series to A1, and the ratio to Q.

(1) S (n) = A1 + A1 * q + A1 * q ^ 2 +... + A1 * q ^ (n-1 );
(2) S (n + 1) = A1 + A1 * q + A1 * q ^ 2 + .... + A1 * q ^ (n-1) + A1 * q ^ N;
Obtained by (2) minus (1)
(3) S (n + 1)-S (n) = A1 * q ^ N;
Obtained by S (n) * q
(4) S (n) * q = A1 * q + A1 * q ^ 2 +... + A1 * q ^ N;

Obtained by (2)-(4)
(5) S (n + 1)-S (n) * q = A1;

Obtained by (3)-(5)
[S (n + 1)-S (n)]-[S (n + 1)-S (n) * q] = A1 * q ^ N-A1;
Simplified:
S (n) * q-S (n) = A1 * (Q ^ n-1 );
That is, S (n) = A1 * (Q ^ n-1)/(Q-1 );