Question

Suppose that the sequence x0, x1, x2... is defined by x0 = 3, x1 = 7,...

Suppose that the sequence x0, x1, x2... is defined by x0 = 3, x1 = 7, and xk+2 = xk+1+20xk for k?0. Find a general formula for xk.

I don't even know how to start this.

Thanks!

Math   Algebra   
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Homework Answers

Answer #1

Solution:

Since   is a linear homogeneous recurrence , the characteristic equation is

Divide both side by we get

So ,  by the Theorem :   r is the solution of   if and only if   is the solution of   we have  ,

   and   are the solution to the given recurrence.

by principle of superposition any linear combination of the solutions is also a solution we have

is also a solution.

Now , using the initial conditions    and   we get

On solving simultaneously we get

The general formula is

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