Ak- 7a(k-1) + 10 a(k-2) = 0 solve recurrence relation

The given recurrence relation is:

\[
A_k - 7A_{k-1} + 10A_{k-2} = 0
\]

This is a second-order linear homogeneous recurrence relation with constant coefficients. To solve this, we follow these steps:

### Step 1: Set up the characteristic equation.
Assume a solution of the form \( A_k = r^k \), where \( r \) is some constant. Substituting this into the recurrence relation:

\[
r^k - 7r^{k-1} + 10r^{k-2} = 0
\]

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

\[
r^2 - 7r + 10 = 0
\]

### Step 2: Solve the characteristic equation.
The characteristic equation is:

\[
r^2 - 7r + 10 = 0
\]

Solve this quadratic equation using the quadratic formula:

\[
r = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(10)}}{2(1)}
\]
\[
r = \frac{7 \pm \sqrt{49 - 40}}{2}
\]
\[
r = \frac{7 \pm \sqrt{9}}{2}
\]
\[
r = \frac{7 \pm 3}{2}
\]

So, the roots are:

\[
r_1 = \frac{7 + 3}{2} = 5, \quad r_2 = \frac{7 - 3}{2} = 2
\]

### Step 3: General solution.
Since the characteristic equation has two distinct real roots \( r_1 = 5 \) and \( r_2 = 2 \), the general solution to the recurrence relation is:

\[
A_k = C_1 \cdot 5^k + C_2 \cdot 2^k
\]

where \( C_1 \) and \( C_2 \) are constants that can be determined from initial conditions.

If initial conditions are provided (such as values of \( A_0 \) and \( A_1 \)), we can solve for \( C_1 \) and \( C_2 \).

Would you like to provide initial conditions, or do you need further details?

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Here are 5 related questions:
1. How do initial conditions affect the solution of a recurrence relation?
2. What happens if the characteristic equation has repeated roots?
3. Can this method be applied to non-homogeneous recurrence relations?
4. How does solving a recurrence relation differ from solving a differential equation?
5. What are some common applications of recurrence relations in computer science?

**Tip:** When solving recurrence relations, always check for distinct, repeated, or complex roots in the characteristic equation.

Ak - 7a(k-1) + 10 a(k-2) = 0 solve recurrence relation

Learn how to solve the recurrence relation Ak - 7A(k-1) + 10A(k-2) = 0 using the characteristic equation method. This solution involves solving a quadratic equation and finding the general solution for distinct real roots. Suitable for students in Grades 11-12 and early college.

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