Answer
See explanation
Work Step by Step
Below is the core reason why the proposed definition fails to produce a well‐defined function. The “definition” is
\[
G(n) \;=\;
\begin{cases}
1, & \text{if }n=1,\\[4pt]
G\!\bigl(\tfrac{n}{2}\bigr), & \text{if }n\text{ is even},\\[4pt]
2 + G(3n - 5), & \text{if }n\text{ is odd and }n>1.
\end{cases}
\]
We want to see that there is **no** consistent way to assign values to \(G\) for *all* \(n\ge1\).
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## Key Contradiction at \(n=5\)
1. For \(n=5\) (which is odd and \(>1\)), the rule says:
\[
G(5) \;=\; 2 + G\bigl(3\cdot 5 - 5\bigr)
\;=\; 2 + G(10).
\]
2. But \(10\) is even, so:
\[
G(10) \;=\; G\!\bigl(\tfrac{10}{2}\bigr)
\;=\; G(5).
\]
3. Combining these:
\[
G(5)
\;=\; 2 + G(10)
\;=\; 2 + G(5).
\]
This implies \(0 = 2\), a direct contradiction.
Thus, as soon as we try to define \(G(5)\), we find it depends on \(G(10)\), but \(G(10)\) in turn depends back on \(G(5)\), forcing \(G(5)=2+G(5)\). No single integer value for \(G(5)\) can satisfy that equation.
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## Conclusion
A properly defined function must assign exactly one well‐defined value to each input in its domain. Here, the attempt to define \(G\) leads to a logical impossibility for \(n=5\). Therefore, **the “definition” of \(G\) is not well‐defined**—Student D’s claim is justified.
