Math Problem Statement
Substitutionsmethode anwenden:
Substitutionsmethode anwenden für die Rekurrenzrelation: T_4(n) = { 3522 falls n = 1, 2 * T_4(n - n/4) + n^3 sonst }
Solution
To analyze this recurrence relation using the substitution method, let's start by understanding and solving it step-by-step.
The recurrence relation provided is:
\begin{cases} 3522 & \text{if } n = 1, \\ 2 \cdot T_4\left(n - \frac{n}{4}\right) + n^3 & \text{otherwise}. \end{cases}$$ ### Steps for Solution: 1. **Base Case Analysis**: The base case $$T_4(1) = 3522$$. 2. **Recursive Expansion**: For the recursive case $$T_4(n) = 2 \cdot T_4\left(n - \frac{n}{4}\right) + n^3$$, we need to expand this further to understand how the terms grow. 3. **Rewriting the Recursive Argument**: Notice that in each recursive step, $$n$$ is reduced by $$\frac{n}{4}$$, which can be rewritten as: $$n_{\text{new}} = n - \frac{n}{4} = \frac{3n}{4}.$$ Thus, each recursive call reduces $$n$$ by a factor of $$\frac{3}{4}$$. 4. **Substitute Successively**: Let’s substitute the recurrence to see how $$T_4(n)$$ unfolds. For example: - First substitution: $$T_4(n) = 2 \cdot T_4\left(\frac{3n}{4}\right) + n^3.$$ - Second substitution: $$T_4(n) = 2 \left(2 \cdot T_4\left(\frac{3^2 n}{4^2}\right) + \left(\frac{3n}{4}\right)^3\right) + n^3.$$ - Continuing this pattern, we will reach a point where $$n$$ becomes close to 1, and we can then apply the base case. 5. **Guess a Solution Form**: Based on the recurrence structure, we can guess that $$T_4(n)$$ has a form that involves $$n^3$$ due to the $$n^3$$ term in each recursive call. 6. **Verification by Substitution**: Use induction to verify the guessed form by substituting it back into the recurrence. Would you like to continue with further detailed steps, or do you have specific questions about any part of this process? --- Here are five questions that might help deepen understanding: 1. How does changing the recursive factor (like using $$\frac{3}{4}$$) affect the solution to a recurrence relation? 2. What would the recurrence look like if the term $$n^3$$ were changed to $$n^2$$ instead? 3. How can we determine the growth rate of a recurrence relation without fully solving it? 4. Why is it important to consider the base case when analyzing recurrence relations? 5. How does the method of repeated substitution differ from other methods like the Master Theorem? **Tip:** When solving recurrence relations, try to observe patterns in the recursive reductions; it often reveals the solution's form without full expansion.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Recurrence Relations
Substitution Method
Algebra
Formulas
T_4(n) = 2 * T_4(n - n/4) + n^3
Theorems
Substitution Method for Recurrence Relations
Suitable Grade Level
Undergraduate Level
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