High school series of all general formula method, such as split method, dislocation subtraction method and so on! series

High school series of all general formula method, such as split method, dislocation subtraction method and so on! series

The split term method and the dislocation subtraction method are generally summation methods For general terms, we can refer to the following: (1) first order constant coefficient linear recurrence sequence and undetermined coefficient method a (n + 1) = k * an + H (n ∈ n *, K, h is constant), where when k = 1, {an} is arithmetic sequence, especially when k = 1 and H = 0, {an} is constant sequence, K is not

How to sum the sequence n / 2 ^ n by dislocation subtraction?

Example: find the first n terms of an = n / 2 ^ n and Sn?
Sn=1*1/2^1+2*1/2^2+3*1/2^3+4*1/2^4+.+n/2^n
1/2*Sn=1*1/2^2+2*1/2^3+3*1/2^4+4*1/2^5+.+(n-1)/2^n+n/2^(n+1)
From the above formula to the following formula:
1/2*Sn=(1/2+1/2^2+1/2^3+1/2^4+.1/2^n)-n/2^(n+1)
1/2*Sn=[1/2-1/2^(n+1)]/(1-1/2)-n/2^(n+1)
1/2*Sn=1-2/2^(n+1)-n/2^(n+1)
1/2*Sn=1-(n+2)/2^(n+1)
Sn=2-(n+2)/2^n

Given the square of a-5a-1 = 0, find the square of a + 1 / 2 of a and the value of a + 1 / 2 of A

a²-1=5a
Divide both sides by A
a-1/a=5
Square on both sides
a²-2+1/a²=25
So a & # 178; + 1 / A & # 178; = 27
(a+1/a)²
=a²+2+1/a²
=27+2
=29
So a + 1 / a = ± √ 29