characteristic equation

Question

Apply the method of the characteristic equation to find an explicit closed formula for the numbers an that satisfy the recurrence equation
an = 2an-1(n-1在a下面) - an-2(n-2在a下面)
for n>= 2, with the initial values a0 = 4 and a1 = 1.

Answer 1

Using characteristics equation:
Let an=αn 代入
得到 αn = 2αn-1 - αn-2
除以 αn-2 得 α2 = 2α - 1, 或 (α-1)2 = 0
解為 α = 1, 1 (重根)
所以, an = αn = A1*(1)n + n*A2*(1)n [註: * 是乘]
現在, 用已知的起始值來決定 A1 及 A2 到底是多少:
Let n = 0: ao = A1 + 0 = 4
Let n = 1: a1 = A1 + A2 = 1
上面兩式可知 A1 = 4, A2 = -3
所以, an = 4 - 3n

Answer 2

這不知是工數還是統計問題...希望我翻的對..

應用特徵方程式的方法來求取an的明確閉合公式,其中an必須要能滿足線性迴歸方程式...

這東西早已經還給老師10幾年了..解答就幫不上忙了