Hey guys, just hoping that I could get some help with these 2 questions.

Solve the recurrence relations for these two questions

http://imagehost.platinum.net.au/pic...1bb125fd2d.jpg

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- Jan 14th 2009, 05:16 AMStorm20Recurrence Relation
Hey guys, just hoping that I could get some help with these 2 questions.

Solve the recurrence relations for these two questions

http://imagehost.platinum.net.au/pic...1bb125fd2d.jpg - Jan 14th 2009, 05:20 AMJester
I just posted this

http://www.mathhelpforum.com/math-he...-sequence.html

- it might help.

Added note. In the case of repeated roots of the characteristic equation, solutions are of the form

$\displaystyle a_n = \left( c_1 n + c_2 \right) \rho^n$ - Jan 14th 2009, 12:10 PMStorm20
Yeah I thought it might have had something to do with using the general form, but im not 100 percent sure where or how to use it.!

- Jan 14th 2009, 12:23 PMJester
Let's try the first one. Substitute $\displaystyle a_n = c \rho^n$ into

$\displaystyle a_n - 7 a_{n-1} + 10 a_{n-2} = 0$

gives

$\displaystyle c \rho^n - 7 c \rho^{n-1} + 10 c \rho^{n-2} = 0$

or

$\displaystyle (\rho^2 - 7 \rho + 10) c \rho^{n-2} = 0$

which gives

$\displaystyle (\rho^2 - 7 \rho + 10) = 0$ or $\displaystyle (\rho - 2) \rho - 5) = 0$ so $\displaystyle \rho = 2,\; 5$

So, the general solution to the difference equation is

$\displaystyle a_n = c_1 2 ^n + c_2 5^n$

Now the initial condition

$\displaystyle a_0 = c_1 + c_2 = -1$

$\displaystyle a_1 = c_1 2 + c_2 5 = 4$

Two equations for $\displaystyle c_1\; \text{and}\; c_2$. SOlving gives

$\displaystyle c_1 = -3,\;\;\;c_2 = 2$

leading to the solution

$\displaystyle a_n = -3 \cdot 2 ^n + 2 \cdot 5^n$ - Jan 14th 2009, 06:15 PMStorm20