Solution to Find the solution of the recurrence relation: xn=3xn-1 + 1, where, x0=4 - Sikademy
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Find the solution of the recurrence relation: xn=3xn-1 + 1, where, x0=4

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Let us find the solution of the recurrence relation: x_n=3x_{n-1} + 1, where x_0=4.

For this firstly, let us solve the characteristic equation of the homogeneous equation x_n-3x_{n-1}=0 :

k-3=0 or k=3. It follows that the particular solution of the equation is x_n=a=const, where a=3a+1, and hence a=-\frac{1}{2}. The solution of the recurrence relation: x_n=3x_{n-1} + 1

is x_n=C\cdot3^n-\frac{1}{2}. Taking into account that x_0=4, we conclude that 4=x_0=C-\frac{1}{2}, and thus C=\frac{9}{2}. Consequently,

x_n=\frac{9}{2}\cdot3^n-\frac{1}{2}=\frac{3^{n+2}-1}{2}.


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