Solve the RR an=20an-1 -25an-2 for n>=2 where a0=0 and a1=1. Solve correctly and properly

To solve the recurrence relation \( a_n = 20a_{n-1} - 25a_{n-2} \) for \( n \geq 2 \) with initial conditions \( a_0 = 0 \) and \( a_1 = 1 \), we can follow these steps:

### Step 1: Solve the characteristic equation
The characteristic equation associated with the recurrence relation is obtained by assuming a solution of the form \( a_n = r^n \). Substituting this into the recurrence relation gives:

\[
r^n = 20r^{n-1} - 25r^{n-2}
\]

Dividing through by \( r^{n-2} \) (assuming \( r \neq 0 \)) gives:

\[
r^2 = 20r - 25
\]

This simplifies to the characteristic equation:

\[
r^2 - 20r + 25 = 0
\]

### Step 2: Solve the quadratic equation
We solve the quadratic equation using the quadratic formula:

\[
r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 1 \), \( b = -20 \), and \( c = 25 \):

\[
r = \frac{20 \pm \sqrt{(-20)^2 - 4 \times 1 \times 25}}{2 \times 1}
\]

\[
r = \frac{20 \pm \sqrt{400 - 100}}{2}
\]

\[
r = \frac{20 \pm \sqrt{300}}{2}
\]

\[
r = \frac{20 \pm 10\sqrt{3}}{2}
\]

\[
r = 10 \pm 5\sqrt{3}
\]

So, the roots are \( r_1 = 10 + 5\sqrt{3} \) and \( r_2 = 10 - 5\sqrt{3} \).

### Step 3: Write the general solution
The general solution of the recurrence relation is:

\[
a_n = A(r_1)^n + B(r_2)^n
\]

Substituting the roots:

\[
a_n = A(10 + 5\sqrt{3})^n + B(10 - 5\sqrt{3})^n
\]

### Step 4: Use initial conditions to find \( A \) and \( B \)
We use the initial conditions \( a_0 = 0 \) and \( a_1 = 1 \) to determine \( A \) and \( B \).

For \( n = 0 \):

\[
a_0 = A(10 + 5\sqrt{3})^0 + B(10 - 5\sqrt{3})^0 = A + B = 0
\]

So, \( A + B = 0 \), or \( B = -A \).

For \( n = 1 \):

\[
a_1 = A(10 + 5\sqrt{3}) + B(10 - 5\sqrt{3}) = A(10 + 5\sqrt{3}) - A(10 - 5\sqrt{3})
\]

\[
1 = A[(10 + 5\sqrt{3}) - (10 - 5\sqrt{3})] = A[10 + 5\sqrt{3} - 10 + 5\sqrt{3}] = A[10\sqrt{3}]
\]

\[
1 = 10\sqrt{3} A
\]

\[
A = \frac{1}{10\sqrt{3}}
\]

Since \( B = -A \), we have \( B = -\frac{1}{10\sqrt{3}} \).

### Step 5: Write the final solution
Substituting the values of \( A \) and \( B \):

\[
a_n = \frac{1}{10\sqrt{3}}(10 + 5\sqrt{3})^n - \frac{1}{10\sqrt{3}}(10 - 5\sqrt{3})^n
\]

Simplifying further:

\[
a_n = \frac{1}{10\sqrt{3}} \left[ (10 + 5\sqrt{3})^n - (10 - 5\sqrt{3})^n \right]
\]

This is the closed-form solution for the recurrence relation.

Would you like further details or have any questions?

### Related Questions

1. How do you derive the characteristic equation from a recurrence relation?
2. What are the steps for solving a quadratic characteristic equation?
3. How do you handle complex roots in solving recurrence relations?
4. How can initial conditions change the solution of a recurrence relation?
5. What is the significance of the characteristic roots in a recurrence relation?
6. Can you explain why dividing by \( r^{n-2} \) was valid in this solution?
7. How does one verify the correctness of the obtained closed-form solution?
8. What if the roots of the characteristic equation were repeated?

### Tip
When solving recurrence relations, always check if the characteristic equation has real, distinct, or repeated roots, as this determines the form of the general solution.

Learn how to solve the recurrence relation a_n = 20a_(n-1) - 25a_(n-2) with detailed step-by-step solutions and explanations. This problem covers key concepts in recurrence relations, characteristic equations, and quadratic formulas. Suitable for advanced high school students and beyond.

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