solve the set of recursion equations  a_n = b_{n-1} + c_{n-1}, b_n = a_{n-1}+c_{n-1}, c_n=b_{n-1}+a_{n-1}

We are given a set of recurrence relations:

\[
a_n = b_{n-1} + c_{n-1}
\]
\[
b_n = a_{n-1} + c_{n-1}
\]
\[
c_n = a_{n-1} + b_{n-1}
\]

### Step 1: Analyze Symmetry
All three equations have similar structures. This suggests that the values of \(a_n\), \(b_n\), and \(c_n\) might exhibit a symmetry, so solving one could give insight into the others.

### Step 2: Combine the Equations
Add all three recurrence relations:

\[
a_n + b_n + c_n = (b_{n-1} + c_{n-1}) + (a_{n-1} + c_{n-1}) + (a_{n-1} + b_{n-1})
\]
\[
a_n + b_n + c_n = 2(a_{n-1} + b_{n-1} + c_{n-1})
\]

This simplifies to:
\[
S_n = 2S_{n-1}
\]
where \(S_n = a_n + b_n + c_n\).

### Step 3: Solve the Equation for \(S_n\)
The recurrence relation for \(S_n\) is:
\[
S_n = 2S_{n-1}
\]
This is a geometric progression with the solution:
\[
S_n = S_0 \cdot 2^n
\]
where \(S_0 = a_0 + b_0 + c_0\) is the initial sum of \(a_0\), \(b_0\), and \(c_0\).

### Step 4: Express \(a_n\), \(b_n\), and \(c_n\)
Since the equations are symmetric, we can assume that \(a_n\), \(b_n\), and \(c_n\) grow in a similar manner, and their individual forms can be derived from initial conditions.

For example, if we know initial conditions like \(a_0\), \(b_0\), and \(c_0\), we can calculate subsequent values using the recurrence relations.

Do you have specific initial values for \(a_0\), \(b_0\), and \(c_0\)? This would help in fully determining the individual sequences.

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Do you want more details or have any other questions?

Here are five questions to expand this:

1. How does the solution change if \(a_0 = b_0 = c_0 = 1\)?
2. Can we express each sequence individually if we modify one of the recurrence relations?
3. What happens if one of the sequences is initialized to zero?
4. How does this system behave asymptotically for large \(n\)?
5. What are the consequences of having non-equal initial conditions?

**Tip:** When working with recursive sequences, solving for the sum often simplifies the individual sequence relationships.

Solve the set of recursion equations: a_n = b_{n-1} + c_{n-1}, b_n = a_{n-1} + c_{n-1}, c_n = a_{n-1} + b_{n-1}.

Explore how to solve a set of recursive equations involving three sequences a_n, b_n, and c_n. This solution covers geometric progression and symmetry in recursive sequences. Ideal for students in Grades 10-12.

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