Start saving money today, one on the first day, two on the second day, three on the third day and four on the fourth day. . . . . On the 730th day, this topic seems very difficult. In fact, it can be simply calculated by a mathematical calculation method, which is geometric series.

Original formula = (1+730)+(2+729)+(3+728)++(365+366).

=73 1×730/2

=73 1×365

=2668 15

Expand one's knowledge

Geometric series refers to a series in which the ratio of each term to its previous term is equal to the same constant from the second term, usually expressed by G and P. This constant is called the common ratio of geometric series, usually expressed by the letter Q (q≠0), and geometric series a 1≠ 0. Where each item in {an} is not 0. Note: When q= 1, an is a constant series.

(1) Definition:

(2) General term formula (the general term formula of geometric series is derived from defined multiplication):

(3) Sum formula:

The sum formula is described in words as follows: Sn= the first term (the n power of common ratio 1)1-common ratio (common ratio ≠ 1) If common ratio q= 1, then every term in the geometric series is equal, and the general term formula is, any two terms. What is the relationship? ; When using the sum of the first n terms of geometric series, we must pay attention to discuss whether the common ratio q is 1.

(4) From the definition of geometric series, the general term formula and the first n terms formula, we can deduce that:

(5) Equal comparison:

If, then? For what? Equal ratio mean term.

Remember π n = A 1 A2? An has π2n- 1=(an)2n- 1, π 2n+1= (an+1) 2n+1.

In addition, each term is a geometric series with positive numbers, and 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. In this sense, we say that a positive geometric series and an arithmetic series are isomorphic.

Definition of proportional terms: Starting from the second term, every term (except the last term with finite series) is an item whose previous term is proportional to its subsequent term.

Equal ratio median formula:

or

(6) Infinitely recursive geometric series summation formula:

Sum formula of infinite recursive equal ratio series: the absolute value of common ratio is less than 1, and the limit when n increases infinitely is called the sum of infinite geometric progression terms.

(7) Common ratio of a new geometric series composed of geometric series: {an} is a geometric series with a common ratio of q.

1. If A=a 1+a2++an

B=an+ 1++a2n

C=a2n+ 1+a3n

Then, a, b and c form a new geometric series with the common ratio q = q n.

2. If A=a 1+a4+a7++a3n-2.

B=a2+a5+a8++a3n- 1

C=a3+a6+a9++a3n

Then, a, b and c form a new geometric series, and the common ratio is q = q