What are the summation formulas of equal ratio series?
Geometric series: a (n+ 1)/an = q, where n is a natural number.
(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-a 1q^n)/( 1-q)
= a1/(1-q)-a1/(1-q) * q n (that is, a-AQ n)
(Premise: Q is not equal to 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.
(5) "G is the proportional average 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, a n stands for the n power of a.
Extended reading: how to deduce the summation formula of equal proportion series
A 1 first item, common ratio q
a(n+ 1)=an*q=a 1*q^(n)
Sn=a 1+a2+..+ An
q*Sn=a2+a3+...+a(n+ 1)
qSn-Sn=a(n+ 1)-a 1
s=a 1(q^n- 1)/(q- 1)
1, the meaning of geometric series: a series, if the ratio of any last term to the previous term is the same constant, that is: A(n+ 1)/A(n)=q (n∈N*), this series is called geometric series, where the constant q is called common ratio. For example, 2,4,8, 16...2 10 is a geometric series with a common ratio of 2, which can be written as the square of (A2) =(A 1)x(A3).
2. Sum formula
Sum formula of proportional series: Sn=n×a 1 (q= 1)
sn=a 1( 1-q^n)/( 1-q)=(a 1-an * q)/( 1-q)(q≠ 1)=a 1(q^n- 1)/(q- 1)
(q is the common ratio and n is the number of terms)
Derivation of summation formula of equal proportion sequence;
Sn=a 1+a2+a3+...+an (common ratio q)
q*Sn=a 1*q+a2*q+a3*q+...+an*q=a2+a3+a4+...+a(n+ 1)
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-an*q)/( 1-q)
sn=a 1( 1-q^n)/( 1-q)
3. Mathematics: Mathematics is a discipline that studies concepts such as quantity, structure, change, space and information, and belongs to a formal science from a certain angle. In the words of A Brief History of Mathematics, mathematics is a science that studies various structures (relationships) on sets. It can be seen that mathematics is an abstract subject, and rigorous process is the key to mathematical abstraction. Mathematics plays an irreplaceable role in the development of human history and social life, and it is also an indispensable basic tool for studying and studying modern science and technology.