Solving a Specific Recurrence Relation Using Initial Conditions

Recurrence relations are essential in solving a wide range of problems in mathematics and computer science. In this article, we will delve into the detailed process of solving the recurrence relation (a_n 18a_{n-1} - 36a_{n-2} 3n cdot 9^n) with initial conditions (a_0 4) and (a_1 10).

Step 1: Homogeneous Part

The first step is to solve the homogeneous part of the recurrence relation, which in this case is:

(a_n - 18a_{n-1} 36a_{n-2} 0)

The characteristic equation for this homogeneous part is:

(lambda^2 - 18lambda 36 0)

Solving for (lambda) gives us:

(lambda 9 pm 3sqrt{5})

Thus, the solution to the homogeneous part is:

(a_n C_1 (9 - 3sqrt{5})^n C_2 (9 3sqrt{5})^n)

Step 2: Nonhomogeneous Part

The next step is to find a particular solution to the nonhomogeneous part, which is:

(3n cdot 9^n)

A particular solution of this form can be guessed as:

(v_n (A Bn) 9^n)

Substitution and Solving for Coefficients

Substituting the guessed form into the recurrence relation, we get:

((A Bn) 9^n - 18(A B(n-1)) 9^{n-1} 36(A B(n-2)) 9^{n-2} 3n cdot 9^n)

Dividing through by (9^{n-2}) gives:

((4A 9B(n-1) 81Bn) 9^2 3n cdot 9^2)

Dividing through by (9^2) gives:

((4A 9Bn - 9B 81Bn) 3n)

Rearranging gives:

((85B - 9B)n 4A - 9B 3n)

Matching the coefficients gives us the system:

(85B - 9B 3) and (4A - 9B 0)

Solving these, we get:

(B -frac{27}{5})

(A -frac{54}{5})

Therefore, the particular solution is:

(v_n -frac{54 - 54n}{5} 9^n -frac{3^{2n 3}}{5} n)

General Solution

The general solution for the recurrence relation is the sum of the homogeneous and particular solutions:

(a_n C_1 (9 - 3sqrt{5})^n C_2 (9 3sqrt{5})^n - frac{3^{2n 3}}{5} n)

Initial Conditions

To find the constants (C_1) and (C_2), we use the initial conditions:

(a_0 4) and (a_1 10)

Substituting (n 0) and (n 1) into the general solution gives the system:

(C_1 C_2 4)

((9 - 3sqrt{5}) C_1 (9 3sqrt{5}) C_2 - frac{162}{5} 10)

Solving this system, we get:

(C_1 frac{37}{5} - frac{113sqrt{5}}{150})

(C_2 frac{37}{5} frac{113sqrt{5}}{150})

Final Solution

The final solution to the recurrence relation is:

(a_n left(frac{37}{5} - frac{113sqrt{5}}{150}right) (9 - 3sqrt{5})^n left(frac{37}{5} frac{113sqrt{5}}{150}right) (9 3sqrt{5})^n - frac{3^{2n 3}}{5} n)

Conclusion

The detailed process of solving recurrence relations involves breaking down the problem into a homogeneous and nonhomogeneous part. Each part requires a different approach: finding the characteristic equation for the homogeneous part and guessing a form for the particular solution of the nonhomogeneous part. By combining these solutions and applying initial conditions, we can determine the constants and obtain the final solution.

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