What is the characteristic equation? What are the roots of the characteristic equation?

Understand the Problem

The question is asking to determine the characteristic equation and its roots for a given recurrence relation. The approach involves rewriting the recurrence relation as a characteristic polynomial and solving it to find the roots.

Answer

Characteristic equation: $r^2 - 4r + 3 = 0$; Roots: $r_1 = 3$, $r_2 = 1$.

Steps to Solve

Identify the recurrence relation

The given recurrence relation is

$$ F(n) = 4F(n-1) - 3F(n-2) $$

Formulate the characteristic equation

To form the characteristic equation, replace $F(n)$ with $r^n$, $F(n-1)$ with $r^{n-1}$, and $F(n-2)$ with $r^{n-2}$.

This gives us:

$$ r^n = 4r^{n-1} - 3r^{n-2} $$

Dividing through by $r^{n-2}$ (assuming $r \neq 0$), we obtain:

$$ r^2 = 4r - 3 $$

Rearranging to standard form

Rearranging this equation leads to:

$$ r^2 - 4r + 3 = 0 $$

This is the characteristic equation.

Factoring the characteristic equation

Next, we factor the characteristic equation:

$$ (r - 3)(r - 1) = 0 $$

Finding the roots

Setting each factor equal to zero gives:

$r - 3 = 0 \implies r_1 = 3$
$r - 1 = 0 \implies r_2 = 1$

Thus, the roots of the characteristic equation are:

$r_1 = 3$
$r_2 = 1$

The characteristic equation is

$$ r^2 - 4r + 3 = 0 $$

The roots of the characteristic equation are $r_1 = 3$ and $r_2 = 1$.

More Information

The characteristic equation represents the polynomial formed from the recurrence relation that helps in finding the solution to the recurrence. The roots indicate the rate of growth or decay of the sequence defined by the recurrence relation.

Tips

Confusing the terms in the recurrence relation, which can lead to an incorrect characteristic equation.
Forgetting to divide by $r^{n-2}$, which is necessary to derive the characteristic polynomial correctly.
Incorrect factoring of the characteristic polynomial.

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