The formula of proportional sequence is a formula for finding the sum of a certain number of geometric progression in mathematics. In addition, each term is a positive geometric series, and the power of the same base is taken to form a arithmetic progression; On the other hand, taking any positive number c as the cardinal number and a arithmetic progression term as the exponent, a power energy is constructed, which is a geometric series.

The essence of geometric series:

1, in the geometric series {an}{an}, if m+n=p+q=2k(m, n, p, q, k∈N? )m+n=p+q=2k(m，N，p，q，k∈N？ )，am？ an=ap？ aq=a2kam？ an=ap？ aq=ak2 .

2. If the series {an}{an} and {bn}{bn} (with the same number of items) are geometric progression, then {λan}(λ≠0){λan}(λ≠0), {1an} {1an. bn}{an？ Bn}, {anbn}{anbn} or geometric series.

3. In the geometric series {an}{an}, taking several terms at equal intervals also constitutes a geometric series, namely, an, an+k, an+2k, an+3k,? Ann, Ann +k, Ann +2k, Ann +3k,? Is a geometric series, and the common ratio is qkqk.

4.q≠ 1q≠ 1, the 2n2n term of the geometric series with S even number =a2? [ 1? (q2)n] 1？ Q2S even number =a2? [ 1? (q2)n] 1？ Q2, s odd number =a 1? [ 1? (q2)n] 1？ Q2S odd number =a 1? [ 1? (q2)n] 1？ Q2, then S even S odd =qS even S odd = Q.

5. The monotonicity of geometric series depends on the values of two parameters a 1a 1 and qq, an=a 1? qn？ 1an=a 1？ qn？ 1。