Consider this recurrence relation: T(1) = 1 T(2) = 1 T(n) = 4 T(n-2) + 2n2 for n>2 The Master Theorem tells us Group of answer choices

The Master Theorem provides a method for analyzing the time complexity of recurrence relations of the form:

$T(n) = aT\left(\frac{n}{b}\right) + f(n)$

where $a \geq 1$ and $b > 1$. However, the recurrence relation you've given does not fit this form directly since the recursive call is based on $n-2$ rather than dividing $n$.

To analyze this recurrence, we can use substitution or expansion to find a pattern. Here are the steps for doing so:

1. Break Down the Problem
   We have the recurrence: $T(n) = 4T(n-2) + 2n^2 \quad \text{for } n > 2$ with initial conditions $T(1) = 1$ and $T(2) = 1$.

2. Relevant Concepts
   We will expand the recurrence to express $T(n)$ in terms of previous values: $T(n) = 4T(n-2) + 2n^2$ We can substitute $T(n-2)$ based on the same form: $T(n-2) = 4T(n-4) + 2(n-2)^2$

3. Analysis and Detail
   Substituting back into the original equation iteratively:

   • First substitute $T(n-2)$: $T(n) = 4(4T(n-4) + 2(n-2)^2) + 2n^2 = 16T(n-4) + 8(n-2)^2 + 2n^2$
   • Now substitute further: $T(n) = 16(4T(n-6) + 2(n-4)^2) + 8(n-2)^2 + 2n^2$

   Continuing this substitution until reaching the base cases leads to identifying how many terms will contribute to $T(n)$.

4. Verify and Summarize
   We can explore the pattern. Each recursive step effectively adds terms of the form $2k^2$ with a multiplying factor increasing by 4 for each step back. It will be complex without exact forms so setting bounds directly might be easier.

   Ultimately, recognizing that this generates a quadratic sum leads to overall polynomial growth.

Final Answer

The recurrence grows as we recursively apply the relation, indicating a non-linear polynomial time complexity, roughly of the order $O(n^3)$ when approximated through expansion and verifying dominating terms in calculations. However, without a straightforward fit to the Master Theorem or an explicit coefficient identification, it may require deeper methods (like generating functions) for a concrete exact closed form. Thus, identify that growth resembles cubic behavior due to iterated summing of quadratic contributions.