A(n+2)=-A(n+ 1)+2An

(The brackets after A represent subscripts) Find the general term of A..

At that time, I remembered a method of this body: after the original deformation,

A(n+2)+A(n+ 1)-2An=0

manufacture

X^2+X-2=0

The solution is X=-2.

or

1

Therefore, {a (n+ 1)-an} is a sequence with a public ratio of -2; {a (n+ 1)+2an} is a sequence with the public ratio of 1.

Then stand together.

Solve it

The above methods should be said to be characteristic root method and fixed point method.

Characteristic root:

For the recurrence formula of multiple continuous terms (excluding constant terms), it can be simplified to the (n- 1) order equation of X. 。

Namely: A0 * an+a1* an+1+A2 * an+2+... AK * an+k can be written as:

a0+a 1x+a2x^2+...akx^(k- 1)=0

Then find the roots (both real roots and imaginary roots are acceptable). Different terms are written as c * x (n- 1), and the same term is written as an algebraic expression about n. How many roots are there? The degree of n is the number of roots minus 1. For example, find x 1 = 2, x2 = 3, x3 = 3 and x4 = 6.

Fixed point:

For example, a 1= 1, a(n+ 1)= 1+2/an are known.

(n is greater than or equal to 1), find one.

a(n+ 1)=(an+2)/an(*)

Let an = x and a (n+1) = x.

x=(x+2)/x

x^2-x-2=0

x 1=2，x2=- 1

{(an-2)/(an+ 1)} is a geometric series.

Order (an-2)/(an+ 1)=bn

b(n+ 1)/bn =[(a(n+ 1)-2)/(a(n+ 1)+ 1)]/[(an-2)/(an+ 1)]

(replace a with an (n+1)

=- 1/2

b(n+ 1)=(- 1/2)bn

b 1=- 1/2

bn=(- 1/2)^n=(an-2)/(an+ 1)

an=[2+(- 1/2)^n]/[ 1-(- 1/2)^n],n>； = 1

Note: Fractional recursion in the form of a(n+ 1)=(Aan+B)/(Can+D), where a and c are not 0, can be solved by fixed point method. Let a(n+ 1)=an=x and substitute it into the quadratic equation about x.

(1) If two x 1 are not equal to x2, then one {(an-x 1)/(an-x2)} is a geometric series, and the common ratio is obtained by two quotients.

(2) If two x 1 equals x2, {1/(an-x 1)} is arithmetic progression, the tolerance is calculated by the difference between the two terms.

If there is no solution, we must find another way.

Moreover, the fixed point is generally only used when the score is up and down once, and it will not work if there is a second possibility.

For the principle, it needs a university to learn, which is based on the study of equations.