2. (a) Use the master theorem to find the exact solution of the following recurrence equation. Make sure you find the constants. Assume n is a power of 2. +n, n ≥ 2 4T 3, n = 1 T(n) = 2(b) Use repeated substitution to find the exact solution of the following recurrence. T(n) = { 1, n+T(n-1), n ≥2 n = 1

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Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
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can you solve 2020 image one I also provided refrences how MY prof solved so based on that pleease solve this image 2020 one step by step formet please use from refrnecs 

**Problem Statement:**
2. Use the master theorem to find the exact solution of the following recurrence equation.  
Find the constants. Assume \( n \) is an integer power of 2, \( n = 2^k \).

\[
T(n) = 
\begin{cases} 
27 \left(\frac{n}{2}\right) + n^2, & n \geq 2 \\
1, & n = 1 
\end{cases}
\]

**Solution Using Master Theorem:**

1. **Master Theorem Constants:**
   - \( a = 2 \), \( b = 2 \), \( \beta = 2 \)
   - \( h = \log_b a = 1 \)

   Since \( h \neq \beta \), we proceed with:

   \[
   T(n) = A n^h + B n^\beta
   \]

   Substituting the values for \( h \) and \( \beta \):

   \[
   T(n) = A n + B n^2
   \]

2. **Finding Constants \( A \) and \( B \):**

   - \( T(1) = 1 = A + B \)
   - \( T(2) = 2T(1) + 2^2 = 2 + 4 = 6 = 2A + 4B \)

   From the equations:
   - \( A + B = 1 \)
   - \( 2A + 4B = 6 \)

   Solving the equations:
   - Rearrange: \( 2B = 4 \) implies \( B = 2 \)
   - Substitute: \( A + 2 = 1 \) implies \( A = -1 \)

3. **Final Solution:**

   Substitute the constants back into the equation:

   \[
   T(n) = 2n^2 - n
   \]

This gives the exact solution to the recurrence using the master theorem.
Transcribed Image Text:**Problem Statement:** 2. Use the master theorem to find the exact solution of the following recurrence equation. Find the constants. Assume \( n \) is an integer power of 2, \( n = 2^k \). \[ T(n) = \begin{cases} 27 \left(\frac{n}{2}\right) + n^2, & n \geq 2 \\ 1, & n = 1 \end{cases} \] **Solution Using Master Theorem:** 1. **Master Theorem Constants:** - \( a = 2 \), \( b = 2 \), \( \beta = 2 \) - \( h = \log_b a = 1 \) Since \( h \neq \beta \), we proceed with: \[ T(n) = A n^h + B n^\beta \] Substituting the values for \( h \) and \( \beta \): \[ T(n) = A n + B n^2 \] 2. **Finding Constants \( A \) and \( B \):** - \( T(1) = 1 = A + B \) - \( T(2) = 2T(1) + 2^2 = 2 + 4 = 6 = 2A + 4B \) From the equations: - \( A + B = 1 \) - \( 2A + 4B = 6 \) Solving the equations: - Rearrange: \( 2B = 4 \) implies \( B = 2 \) - Substitute: \( A + 2 = 1 \) implies \( A = -1 \) 3. **Final Solution:** Substitute the constants back into the equation: \[ T(n) = 2n^2 - n \] This gives the exact solution to the recurrence using the master theorem.
***Educational Resource: Master Theorem and Repeated Substitution for Recurrence Relations***

**Problem 2: Solving Recurrence Relations**

**2(a) Using the Master Theorem**

To find the exact solution of the following recurrence equation, apply the Master Theorem. Ensure the constants are identified. Assume \( n \) is a power of 2.

\[
T(n) =
\begin{cases} 
4T\left(\frac{n}{2}\right) + n, & n \geq 2 \\ 
3, & n = 1 
\end{cases}
\]

**2(b) Using Repeated Substitution**

Employ repeated substitution to find the exact solution for the following recurrence.

\[
T(n) =
\begin{cases} 
n + T(n-1), & n \geq 2 \\ 
1, & n = 1 
\end{cases}
\]

---

Each part provides a distinct approach to solving recurrence relations, offering insight into how recursive algorithms' time complexities can be derived.
Transcribed Image Text:***Educational Resource: Master Theorem and Repeated Substitution for Recurrence Relations*** **Problem 2: Solving Recurrence Relations** **2(a) Using the Master Theorem** To find the exact solution of the following recurrence equation, apply the Master Theorem. Ensure the constants are identified. Assume \( n \) is a power of 2. \[ T(n) = \begin{cases} 4T\left(\frac{n}{2}\right) + n, & n \geq 2 \\ 3, & n = 1 \end{cases} \] **2(b) Using Repeated Substitution** Employ repeated substitution to find the exact solution for the following recurrence. \[ T(n) = \begin{cases} n + T(n-1), & n \geq 2 \\ 1, & n = 1 \end{cases} \] --- Each part provides a distinct approach to solving recurrence relations, offering insight into how recursive algorithms' time complexities can be derived.
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