What is the proportional sequence formula, what is the proportional sequence formula all

1) The general formula for proportional series is: an=a1 q (n 1).

If the general term formula is deformed to an=a1 q*q n(n n*), then at q 0, an can be regarded as a function of the independent variable n, and the point (n,an) is a group of isolated points on the curve y=a1 q*q x.

The relation between any two am, an is an=am·q (n-m).

3) From the definition of the proportional series, the formula of the general term, the first n term and the formula can be deduced:

a1·an=a2·an-1=a3·an-2=…=ak·an-k+1，k∈

4) Proportional middle term: aq·ap=ar 2, ar is ap, aq proportional middle term.

Note n=a1·a2....an, then there is 2n-1=(an)2n-1, 2n+1=(an+1)2n+1

In addition, a proportional series where all items are positive numbers is taken from the same base number to form an equal difference series; Conversely, if you take any positive number c as the base and use the terms of an equal difference series as an exponential power to construct can, it is an equal proportional series. In this sense, we say that a positive proportional series is "isomorphic" to a difference series.

Nature: If. m, n, p, q n*, and m n=p q, then am·an=ap·aq;

In the proportional series, each in turn.

The sum of k terms is still an equal proportional sequence.

g is the proportional middle term of a and b""g 2=ab(g≠0)".

The sum of the first n terms of the proportional series sn=a1(1-q n) (1-q) or sn=(a1-an*q) (1-q)(q≠1).

sn=n*a1

q=1) In a proportional series, neither the first term a1 nor the common ratio q are zero.

Note: In the above formula, a n denotes a to the nth power.

Equal difference series and formulas.

sn=n(a1+an) 2=na1+n(n-1) 2d proportional sequence summation formula.

Q≠1. sn=a1(1-q^n)/(1-q)=(a1-anq)/(1-q)

Q=1 when sn=na1

A1 is the first term, An is the nth term, D is the tolerance, Q

is proportional).

Proportional series full formula:

1) The general formula for proportional series.

Yes: an=a1 q (n 1).

If the general term formula is deformed to an=a1 q*q n(n n*), when q>0, an can be regarded as an independent variable.

As a function of n, the point (n,an) is a group of isolated dot bands and sails on the curved shed megaline y=a1 q*q x.

2) The relation between any two terms am, an is an=am·q (n-m).

3) From the definition of the proportional series, the formula for the general term, the first n terms and the formula, it can be deduced: a1·an=a2·an-1=a3·an-2=....=ak·an-k+1，k∈。

4) Proportional terms.

aq·ap=ar 2, ar is the middle term of ap, aq and equal ratio.

5) Proportional summation: sn=a1+a2+a3+.an。

When Q≠1, sn=a1(1-q n) (1-q) or sn=(a1-an q) (1-q).

When q=1, sn=n a1(q=1).

Note n=a1·a2....an, then there is 2n-1=(an)2n-1, 2n+1=(an+1)2n+1

Formula: Q≠1, sn=a1(1-q n) (1-q)=(a1-anq) (1-q). Q=1 when sn=na1

A1 is the first term, An is the nth term, and Q is the proportional term).

Geometric progression. It refers to a series of cracks from the second term onwards, in which the ratio of each term to its predecessor is equal to the same constant, which is often represented by g and p. This constant is called the common ratio of the proportional series, which is usually represented by the letter q (q≠0), and the proportional series a1≠ 0.

Special properties: If m, n, p, q n, and m+n=p+q, then am an=ap aq.

In the proportional series, the sum of each k terms is still a proportional series; The special properties of proportional series.

If m, n, q n, and m+n=2q, then am an=(aq) 2.

If g is the proportional middle term of a and b.

then g 2 = ab (g ≠ 0).

In the proportional series, the first Qingyuan union a1 and the common ratio q are not zero.

Note: In the above formula, an denotes the nth term of the proportional series.

Proportional series full formula:

1) The general formula for proportional series.

Yes: an=a1 q (n 1).

If the general term formula is deformed to an=a1 q*q n(n n*), when q>0, an can be regarded as an independent variable.

n, the point (n,an) is the isolated point of a group of dust and silver hail on the curve y=a1 q*q x.

2) For any two AM, AN is AN=AM·q (N-M).

3) From the definition of the proportional series, the general term, the first n terms and the formula, it can be deduced: a1·an=a2·an-1=a3·an-2=....=ak·an-k+1，k∈。

Derivation of the summation formula:

1）sn=a1+a2+a3+..an (the common ratio is q).

2) Paifon qsn=a1q + a2q + a3q +anq = a2+ a3+ a4+..an+ a(n+1)

3）sn-qsn=(1-q)sn=a1-a(n+1)

4）a(n+1)=a1qn

5）sn=a1(1-qn)/(1-q)（q≠1)

The formula for the proportional difference series is shown below:

A proportional sequence formula is a mathematical formula for finding the sum of a certain number of proportional sequences. In addition, a proportional series where all terms are positive numbers are formed by taking the same base exponent to form an equal difference series; Conversely, if you take any positive number c as the base and use the terms of an equal difference series as an exponential power to construct can, it is an equal proportional series.

The nature of the proportional series:

1. In the proportional series, if m+n=p+q=2k(m,n,p,qingdong q,k n)m+n=p+q=2k(m,n,p,q,k n), then am an=ap aq=a2kam an=ap aq=ak2.

2. If the number of sequences, (the same number of terms) is proportional to the series, then (≠0) (0), is still the proportional series.

3. In the proportional series, taking out a number of terms at equal distances also constitutes an proportional series, that is, an, an+k, an+2k, an+3k, an,an+k, an+2k, an+3k, which is an equal proportional series, and the common ratio is qkqk.

4. Q≠1q≠1 is the first 2n2n terms of the proportional series, Yuchashan s-even = a2 [1 (q2)n]1 q2s even=a2 [1 (q2)n]1 q2, s odd = a1 [1 (q2)n]1 q2s odd=a1 [1 (q2)n]1 q2, then s even s odd = qs even s odd = q.

5. The monotonicity of the proportional series depends on the values of the two parameters A1A1 and QQ, an=a1 qn 1an=a1 qn 1.

The formula for summing the proportional series is sn=a1(1-q n) (1-q).

1. Common formulas for proportional series.

A proportional sequence is a sequence in which each number in a sequence is equal in proportion to its predecessor. The formula is: an=a1 r (n-1).

where an is the nth term of the series, a1 is the first term of the series, r is the fixed scale factor, and n is the number of terms. The sum of the first n terms of the proportional series is: sn=a1 (1-r n) (1-r).

where sn denotes the sum of the first n terms of the series, a1 is the first term of the series, r is the fixed scale factor, and n is the number of terms. The numerator in this formula is derived from the summation formula of the proportional series, and the sum of the first n terms of the proportional sequence is: sn=a1 (1-r n) 1-r).

To explain simply, the numerator is the result of the addition of the first n terms of the sequence, and the denominator is a fixed value, which is used to ensure that the ratio of the sum of the numerator and the subsequent terms is the same. This formula can conveniently calculate the sum of the first n terms of a proportional series, and it is also one of the commonly used formulas in mathematics.

2. Matters needing attention.

When applying the formula calculation of proportional series, you should first use $a 1$ and $q$ to determine the characteristics of the series, and then find the sum of specific terms or the first n terms as needed. In addition, care needs to be taken to choose the appropriate calculation method and pay attention to the meaning of the parameters in the formula.

Introduction to Proportional Series:

A proportional sequence is a sequence of numbers in which the ratio of two adjacent terms is a fixed constant, which is called the common ratio. Let the first term of the proportional series be a1 and the common ratio be q, then the general form of the series is: a1, a1 q, a1 q 2, a1 q 3, etc.

That is, the first term is a1, and each subsequent term is the previous term multiplied by the common ratio q. Here q can be positive, negative, or zero, and as long as it is not equal to 1, it can form a proportional sequence.

There are some special properties of the proportional series, starting from the second term, the ratios between the adjacent two terms are equal, i.e., a2 a1=a3 a2=a4 a3=.q。From the nth term, the ratio between any two items is equal, i.e., an am=(an-1) a(m-1)=q (n-m).

Proportional sequences are widely used in mathematics, such as calculating compound interest, proportional annual growth rate, proportional scaling, etc. In addition, in the fields of physics, astronomy, ecology and other sciences, proportional sequences are often used to describe the regularity of various natural phenomena.

Proportional sequence summation formula:

1) When Q≠1, sn=a1(1-q n) (1-q)=(a1-anq) (1-q).

2) When Q=1, sn=na1. (a1 is the first term, an is the nth term, q is the same ratio) sn=a1(1-q n) (1-q) The derivation process:

sn=a1+a2+……an

q*sn=a1*q+a2*q+……an*q=a2+a3+……a(n+1)

sn-q*sn=a1-a(n+1)=a1-a1*q^n(1-q)*sn=a1*(1-q^n)

sn=a1*(1-q^n)/(1-q)

(1) Proportional series: a (n+1) an=q (n n).

2) General formula: an=a1 q (n-1);

Promotional: an=am q (n-m);

3) Summation formula: sn=n a1 (q=1)sn=a1(1-q n) (1-q) =(a1-an q) (1-q) (q≠1) (q is the ratio, n is the number of terms).

4) Properties: If m, n, p, q n, and m n=p q, then am an=ap aq;

In a proportional sequence, the sum of each k terms remains in proportional sequence.

If m, n, q n, and m+n=2q, then am an=aq 2(5)."g is the proportional middle term of a and b""g^2=ab（g ≠ 0）".

6) In the proportional series, neither the first term A1 nor the common ratio q are zero.

Note: In the above formula, an denotes the nth term of the proportional series.

Derivation of the summation formula for proportional sequences: sn=a1+a2+a3+.an(common ratio is q) q*sn=a1*q+a2*q+a3*q+..

an*q =a2+a3+a4+..a(n+1)

sn-q*sn=a1-a(n+1)

1-q)sn=a1-a1*q^n

sn=(a1-a1*q^n)/(1-q)

sn=a1(1-q^n)/(1-q)

The number in the n-series For example, 1, 2, 4, 8....At this time, a1=1, the common ratio is 2, and the formula for the same term is an=a1q (n-1), and the first few numbers are required to be substituted into that number, n is that number, understand this, and then look at the derivation of the textbook, I think you will figure it out.

a, b, and c are proportional series.

b square = ac

b a=c b=common ratio.

The first term a1, the common ratio q a(n+1)=an*q=a1*q(n-1) "followed by q-n-1 power".

If there are four proportional sequences of ABCD, then a*d=b*c