Proved by definition, that is, prove an-an- 1=m (constant); Prove by the nature of arithmetic progression, that is, prove that 2an = an-1+an+1; It is proved that there is always an arithmetic mean, that is, 2an = a (n-1)+a (n+1); The sum of the first n terms conforms to sn = an 2+bn.

Arithmetic progression's definition:

Arithmetic progression refers to a series of the same constant whose difference between each term and the previous term is equal to the second term, usually expressed by a and p. This constant is called arithmetic progression's tolerance, and is often expressed by the letter d?

For example: 1, 3, 5, 7, 9...2n- 1. The general formula is: an = a1+(n-1) * D. The first term a 1= 1, and the tolerance d=2. The first n terms and formulas are: sn = a1* n+[n * (n-1) * d]/2 or Sn=[n*(a 1+an)]/2. Note: All the above n are positive integers.

The basic nature of arithmetic progression:

For arithmetic progression with tolerance of D, the sequence obtained by adding 1 to each term is still arithmetic progression, and its tolerance is still D; For arithmetic progression with tolerance d, the sequence obtained by multiplying each term by constant k is still arithmetic progression, and its tolerance is KD; If {an}{bn} is arithmetic progression, {an bn} and {Kan+bn} (k and b are nonzero constants) are also arithmetic progression.

For arbitrary m and n, in arithmetic progression, there are: an = am+(n-m) dm, n∈N+). Especially when m = 1, a more general arithmetic progression general term formula is obtained than arithmetic progression's general term formula. Generally, when m+n=p+qm, n, p, q∈N+), am+an=ap+aq.

Arithmetic progression with tolerance d, from which equidistant items are taken out to form a new series, which is still arithmetic progression, and its tolerance is kd( k is the difference of the number of items taken out); The term AK. AK+M. AK+2m...(k, m ∈ n+) and arithmetic progression in the table below form a arithmetic progression with a tolerance of MD.

In arithmetic progression, starting from the second term, each term (except the last term of a finite series) is the arithmetic average of the two terms before and after it; When the tolerance d > 0, the number in arithmetic progression increases with the increase of the number of terms; When d < 0, the number in arithmetic progression decreases with the decrease of the number of terms; When d = 0, the number in arithmetic progression is equal to a constant.

Practical application of arithmetic progression:

Financial field: arithmetic progression can be used to calculate time deposit, fixed investment, equal principal and interest, etc. Logistics field: arithmetic progression can be used to calculate container loading and unloading efficiency, and can also be used to plan route optimization.

Engineering field: arithmetic progression can be used to calculate the length of steel bars and plates. Geography: arithmetic progression can be used to calculate the change of altitude and the temperature of sea water.

Medical field: arithmetic progression can be used to calculate the dose and metabolism of drugs. Education: arithmetic progression can be used to calculate the progress of study, changes in test scores, etc.