(2) General formula: an = a1× q (n-1);

Generalization: an = am× q (n-m);

(3) Sum formula: Sn=n×a 1(q= 1).

sn = a 1( 1-q n)/( 1-q)=(a 1-an×q)/( 1-q)(q≠ 1)()

(4) nature:

(1) if m, n, p, q∈N, m+n = p+q, then am×an = AP×AQ；;

(2) In geometric series, every k term is added in turn and still becomes a geometric series.

③ If m, n, q∈N and m+n=2q, then am× an = AQ 2.

(5) "G is the equal ratio mean of A and B" and "G 2 = AB (G ≠ 0)".

(6) In geometric series, the first term a 1 and the common ratio q are not zero.

Note: In the above formula, an stands for the nth term of geometric series.

Derivation of summation formula of proportional series: Sn=a 1+a2+a3+...+an (common ratio q) q * sn = a1* q+a2 * q+a3 * q+...+an * q = a2+a3+a4+.

Sn-q*Sn=a 1-a(n+ 1)

( 1-q)sn=a 1-a 1*q^n

sn=(a 1-a 1*q^n)/( 1-q)

sn=a 1( 1-q^n)/( 1-q)