If a series starts from the second term, the difference between each term and its previous term is equal to the same constant. This series is called arithmetic progression, and this constant is called arithmetic progression's tolerance, which is usually represented by the letter D.
Arithmetic progression's general formula is:
an = a 1+(n- 1)d( 1)
The first n terms and formulas are:
Sn=na 1+n(n- 1)d/2 or Sn=n(a 1+an)/2(2).
It can be seen from the formula (1) that an is a linear function (d≠0) or a constant function (d=0) of n, and (n, an) is arranged in a straight line. According to formula (2), Sn is a quadratic function (d≠0) or a linear function (d =
Arithmetic average in arithmetic progression: generally set as Ar, Am+an=2Ar, so Ar is the arithmetic average of Am and An.
The relationship between any two am and an is:
an=am+(n-m)d
It can be regarded as arithmetic progression's generalized general term formula.
From arithmetic progression's definition, general term formula and the first n terms formula, we can also deduce that:
a 1+an = a2+an- 1 = a3+an-2 =…= AK+an-k+ 1，k∈{ 1，2，…，n}
If m, n, p, q∈N*, m+n=p+q, then there is.
am+an=ap+aq
Sm- 1=(2n- 1)an，S2n+ 1 =(2n+ 1)an+ 1
Sk, S2k-Sk, S3k-S2k, …, Snk-S(n- 1)k… or arithmetic progression, and so on.
Sum = (first item+last item) * number of items ÷2
Number of items = (last item-first item) ÷ tolerance+1
First Item =2, Number of Items-Last Item
Last item =2, number of items-first item
Number of items = (last item-first item)/tolerance+1
Arithmetic progression's application:
In daily life, people often use arithmetic progression, for example, to grade the sizes of various products.
When there is little difference between the maximum size and the minimum size, Chang 'an arithmetic progression conducts grading.
If it is arithmetic progression, and there are AP = q and AQ = p, then a (p+q) =-(p+q).
If it is arithmetic progression, and an = m and am = n, then a (m+n) = 0.
Geometric series:
If a series starts from the second term and the ratio of each term to the previous term is equal to the same constant, this series is called geometric series. This constant is called the common ratio of geometric series and is usually represented by the letter Q.
The general formula of geometric series is: an = a 1 * q (n- 1).
(2) the first n terms and formulas are: sn = [a1(1-q n)]/(1-q)
And the relationship between any two terms am and an is an = am q (n-m).
(3) A1an = a2an-1= a3an-2 = … = akan-k+1,k ∈ {1 can be deduced from the definition of geometric series, the general term formula, the first n terms and the formula.
(4) If m, n, p, q∈N*, then: AP AQ = am an.
Mean term of equal ratio: aq ap = 2arar is ap, and mean term of equal ratio of aq.
If π n = A 1 A2 … an, then 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 number 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.
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.
G is the median term in the equal proportion of A and B, and G 2 = AB (G ≠ 0).
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.
Geometric series are often used in life.
For example, banks have a way of paying interest-compound interest.
That is, the previous interest and Hepburn gold price are counted as principal.
In calculating the interest of the next period, which is what people usually call rolling interest.
The formula for calculating the sum of principal and interest according to compound interest: sum of principal and interest = principal *( 1+ interest rate) due.