If a0 = 0, a1 = 1, and ak+1 = 4ak 4ak1 for all k 1, use methods of linear algebra to

Chapter 6, Problem 10

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QUESTION:

If \(a_{0}=0, a_{1}=1 \text {, and } a_{k+1}=4 a_{k}-4 a_{k-1}\) for all \(k \geq 1\), use methods of linear algebra to determine the formula for \(a_k\). (Hint: The matrix will not be diagonalizable, but you can get close if you stare at Exercise 6.2.7.)

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QUESTION:

If \(a_{0}=0, a_{1}=1 \text {, and } a_{k+1}=4 a_{k}-4 a_{k-1}\) for all \(k \geq 1\), use methods of linear algebra to determine the formula for \(a_k\). (Hint: The matrix will not be diagonalizable, but you can get close if you stare at Exercise 6.2.7.)

ANSWER:

Step 1 of 5

Given recurrence relation is

Prescribed initial conditions are

To find the formula for .

Let us define a vector by

Then,

Here,

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