Solve the recurrence relation 𝑎𝑟+2 − 5 𝑎𝑟+1 + 6 𝑎𝑟 = 2 by the method of
generating functions with initial conditions 𝑎0 = 1 and 𝑎1 = 2.

To solve the recurrence relation \( a_{r+2} - 5a_{r+1} + 6a_r = 2 \) using the method of generating functions, we follow these steps:

### 1. **Define the Generating Function**
Let the generating function be:
\[
A(x) = \sum_{r=0}^\infty a_r x^r.
\]

### 2. **Write the Recurrence Relation in Terms of Generating Functions**
Multiply the recurrence relation \( a_{r+2} - 5a_{r+1} + 6a_r = 2 \) by \( x^r \) and sum over \( r \geq 0 \):
\[
\sum_{r=0}^\infty \left( a_{r+2}x^r - 5a_{r+1}x^r + 6a_r x^r \right) = \sum_{r=0}^\infty 2x^r.
\]

### 3. **Shift the Indices to Align Terms**
- For \( \sum_{r=0}^\infty a_{r+2}x^r \), substitute \( r \to r-2 \):
\[
\sum_{r=0}^\infty a_{r+2}x^r = x^2 \sum_{r=0}^\infty a_r x^r = x^2 A(x).
\]
- For \( \sum_{r=0}^\infty a_{r+1}x^r \), substitute \( r \to r-1 \):
\[
\sum_{r=0}^\infty a_{r+1}x^r = x \sum_{r=0}^\infty a_r x^r = x A(x).
\]
- The term \( \sum_{r=0}^\infty a_r x^r = A(x) \) remains unchanged.

Substitute these shifts into the recurrence:
\[
x^2 A(x) - 5x A(x) + 6A(x) = \frac{2}{1-x}.
\]

### 4. **Factorize and Solve for \( A(x) \)**
Group \( A(x) \):
\[
A(x) \left( x^2 - 5x + 6 \right) = \frac{2}{1-x}.
\]
Factorize \( x^2 - 5x + 6 = (x-2)(x-3) \):
\[
A(x) = \frac{2}{(1-x)(x-2)(x-3)}.
\]

### 5. **Partial Fraction Decomposition**
Decompose \( \frac{2}{(1-x)(x-2)(x-3)} \):
\[
\frac{2}{(1-x)(x-2)(x-3)} = \frac{A}{1-x} + \frac{B}{x-2} + \frac{C}{x-3}.
\]
Multiply through by \( (1-x)(x-2)(x-3) \) and solve for \( A, B, C \):
\[
2 = A(x-2)(x-3) + B(1-x)(x-3) + C(1-x)(x-2).
\]
Expanding and equating coefficients gives:
\[
A = -1, \, B = 3, \, C = 0.
\]

Thus:
\[
A(x) = \frac{-1}{1-x} + \frac{3}{x-2}.
\]

### 6. **Expand into Power Series**
Expand each term:
- \( \frac{-1}{1-x} = -\sum_{r=0}^\infty x^r \).
- \( \frac{3}{x-2

Solve the recurrence relation 𝑎𝑟+2 − 5 𝑎𝑟+1 + 6 𝑎𝑟 = 2 by the method of generating functions with initial conditions 𝑎0 = 1 and 𝑎1 = 2.

Discover how to solve the recurrence relation a_(r+2) − 5a_(r+1) + 6a_r = 2 using generating functions. This detailed solution involves recurrence relations, generating function transformations, and partial fraction decomposition. Learn step-by-step with initial conditions a0 = 1 and a1 = 2.

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