Solution Found!
If a0 = 0, a1 = 1, and ak+1 = 4ak 4ak1 for all k 1, use methods of linear algebra to
Chapter 6, Problem 10(choose chapter or problem)
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.)
Questions & Answers
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,