find all solutions of the recurrence relation an=2an-1+2n 2

To find the solutions to the recurrence relation \( a_n = 2a_{n-1} + 2n + 2 \), we can use the method of solving non-homogeneous recurrence relations. Let’s go through the steps:

### Step 1: Identify the Homogeneous Part
The homogeneous part of the recurrence relation is:
\[
a_n^h = 2a_{n-1}^h
\]
This is a first-order linear recurrence relation with the solution of the form:
\[
a_n^h = C \cdot 2^n
\]
where \( C \) is a constant.

### Step 2: Find a Particular Solution
Now we need a particular solution for the non-homogeneous equation:
\[
a_n = 2a_{n-1} + 2n + 2
\]
The non-homogeneous part \( 2n + 2 \) suggests a linear function as a particular solution. Let’s assume \( a_n^p = An + B \) as our particular solution form, where \( A \) and \( B \) are constants to be determined.

Substitute \( a_n^p = An + B \) into the recurrence relation:
\[
An + B = 2(A(n-1) + B) + 2n + 2
\]
Expanding and simplifying,
\[
An + B = 2An - 2A + 2B + 2n + 2
\]
Now, equate the coefficients of \( n \) and the constant terms on both sides:
1. Coefficient of \( n \): \( A = 2A + 2 \Rightarrow -A = 2 \Rightarrow A = -2 \)
2. Constant term: \( B = -2A + 2B + 2 \Rightarrow B = -2(-2) + 2B + 2 \Rightarrow B = 4 + 2 - B \Rightarrow B

Explore the step-by-step solution to the non-homogeneous recurrence relation an = 2an-1 + 2n + 2. This guide covers key mathematical concepts like recurrence relations and linear algebra, using methods to find both homogeneous and particular solutions. Ideal for college-level students and advanced math enthusiasts.

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