xn=30
xn+1 = 1.8xn +15
I'm completely lost with this question, finding the closed form of the linear recurrence sequence
any help would be appreciated, thanks
The statement has no sense. Have you exactly quoted it? .
Fernando Revilla
Hello, entrepreneurforum.co.uk!
$\displaystyle X_{n+1} \:=\:1.8X_n + 15,\;\;X_1 = 30$
Find the closed form of the recurrence sequence.
[rant]
. . . . . Why are these problems assigned to people
. . . . . who have no knowledge of recurrence relations?
[/rant]
$\displaystyle \begin{array}{ccccccc}\text{We have the recurrence: } & X_{n+1} &=& 1.8X_n + 15 & [1] \\
\text{Write the "next" equation:} & X_{n+2} &=& 1.8X_{n+1} + 15 & [2] \end{array}$
Subtract [2] - [1]: .$\displaystyle X_{n+2} - X_{n+1} \;=\;1.8X_{n+1} - 1.8X_n $
. . . . . $\displaystyle X_{n+2} - 2.8X_{n+1} + 1.8X_n \;=\;0$
$\displaystyle \text{Let }Y^n = X_n\!: \;\;Y^{n+2} - 2.8Y^{n+1} + 1.8Y^n \;=\;0$
$\displaystyle \text{Divide by }Y^n\!:\;\;Y^2 - 2.8Y + 1.8 \;=\;0 \quad\Rightarrow\quad (Y - 1)(Y - 1.8)\;=\;0$
. . Hence: .$\displaystyle Y = 1,\;Y = 1.8$
The generating function is of the form: .$\displaystyle f(n) \;=\;A + B(1.8)^n$
We know the first two terms of the sequence: .$\displaystyle f(1) = 30,\;f(2) = 69$
. . $\displaystyle \begin{array}{ccccccc} f(1) = 30\!: & A + 1.8B\; &=& 30 & [3] \\ f(2) = 69\!: & A + 3.24B &=& 69 & [4] \end{array}$
Subtact [4] - [3]: .$\displaystyle 1.44B \:=\:39 \quad\Rightarrow\quad B \:=\:\dfrac{39}{1.44} \:=\:\dfrac{325}{12}$
Substitute into [3]: .$\displaystyle A + 1.8\left(\frac{325}{12}\right) \:=\:30 \quad\Rightarrow\quad A \:=\:\text{-}\dfrac{75}{4}$
Therefore: .$\displaystyle f(n) \;=\;-\dfrac{75}{4} + \dfrac{325}{12}(1.8)^n $
. . . . . . . . $\displaystyle f(n) \;=\;\dfrac{25}{12}\bigg[13(1.8)^n - 9\bigg] $