MHB Solution of the recurrence relation

AI Thread Summary
The discussion focuses on solving the recurrence relation defined by f_n = (2/a^2 + b)f_{n-1} - (1/a^4)f_{n-2} with initial conditions f_0 = 1 and f_{-1} = 0. The user initially calculates several values for f_n but struggles to identify a clear pattern. A suggestion is made to utilize the characteristic equation method, leading to the formulation m^2 - cm - d = 0. The user expresses gratitude for the hint and confirms they have made progress in solving the recurrence relation. The conversation highlights the importance of characteristic equations in solving such mathematical problems.
mathmari
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Hey! :o

How can we solve the following recurrence relation?

$$f_n=\left (\frac{2}{a^2}+b\right )f_{n-1}-\frac{1}{a^4}f_{n-2} \\ f_0=1, f_{-1}=0$$

I calculated some values to see if there is a general pattern, but it doesn't seems so...

$$f_1=\left (\frac{2}{a^2}+b\right ) \\ f_2 =\left (\frac{2}{a^2}+b\right )^2-\frac{1}{a^4} \\ f_3 =\left (\frac{2}{a^2}+b\right )^3-\frac{2}{a^4}\left (\frac{2}{a^2}+b\right ) \\ f_4=\left (\frac{2}{a^2}+b\right )^4-\frac{3}{a^4}\left (\frac{2}{a^2}+b\right ) ^2+\frac{1}{a^8} \\ f_5=\left (\frac{2}{a^2}+b\right )^5-\frac{4}{a^4}\left (\frac{2}{a^2}+b\right ) ^3+\frac{3}{a^8}\left (\frac{2}{a^2}+b\right )$$

Have calculated these values right?

Is there an other way to find the solution of the recurrence relation? (Wondering)
 
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mathmari said:
Hey! :o

How can we solve the following recurrence relation?

$$f_n=\left (\frac{2}{a^2}+b\right )f_{n-1}-\frac{1}{a^4}f_{n-2} \\ f_0=1, f_{-1}=0$$

I calculated some values to see if there is a general pattern, but it doesn't seems so...

$$f_1=\left (\frac{2}{a^2}+b\right ) \\ f_2 =\left (\frac{2}{a^2}+b\right )^2-\frac{1}{a^4} \\ f_3 =\left (\frac{2}{a^2}+b\right )^3-\frac{2}{a^4}\left (\frac{2}{a^2}+b\right ) \\ f_4=\left (\frac{2}{a^2}+b\right )^4-\frac{3}{a^4}\left (\frac{2}{a^2}+b\right ) ^2+\frac{1}{a^8} \\ f_5=\left (\frac{2}{a^2}+b\right )^5-\frac{4}{a^4}\left (\frac{2}{a^2}+b\right ) ^3+\frac{3}{a^8}\left (\frac{2}{a^2}+b\right )$$

Have calculated these values right?

Is there an other way to find the solution of the recurrence relation? (Wondering)
Start with:
[math]f_n = c f_{n-1} + d f_{n-2}[/math]

[math]f_{n + 2} - c f_{n + 1} - d f_n = 0[/math]

The characteristic equation is [math]m^2 - cm - d = 0[/math]

Can you take it from here?

-Dan
 
I did it! Thanks for the hint! (flower)
 
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