Question:
Let \( T(n) \) be the recurrence relation defined as follows:
\[
T(0) = 1, \quad T(1) = 2, \quad T(n) = 5T(n-1) - 6T(n-2) \quad \text{for } n \geq 2
\]
Which one of the following statements is TRUE?
Let \( T(n) \) be the recurrence relation defined as follows:
\[
T(0) = 1, \quad T(1) = 2, \quad T(n) = 5T(n-1) - 6T(n-2) \quad \text{for } n \geq 2
\]
Which one of the following statements is TRUE?
Show Hint
When solving recurrence relations, always find the characteristic equation and analyze the roots to determine the asymptotic behavior.
Updated On: Jan 23, 2025
- \( T(n) = \Theta(2^n) \)
- \( T(n) = \Theta(n2^n) \)
- \( T(n) = \Theta(3^n) \)
- \( T(n) = \Theta(n3^n) \)
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The Correct Option is A
Solution and Explanation
The recurrence relation is:
\[
T(n) = 5T(n-1) - 6T(n-2)
\]
Step 1: Find the characteristic equation.
The characteristic equation is: \[ x^2 - 5x + 6 = 0 \] Factoring gives: \[ (x - 3)(x - 2) = 0 \] Thus, the roots are \( r_1 = 3 \) and \( r_2 = 2 \). Step 2: General solution for the recurrence relation.
The general solution is: \[ T(n) = C_1(3^n) + C_2(2^n) \] Using the initial conditions \( T(0) = 1 \) and \( T(1) = 2 \), we find: \[ C_1 = 0, \quad C_2 \neq 0 \] Step 3: Asymptotic behavior.
Since \( C_1 = 0 \), the solution reduces to: \[ T(n) = C_2(2^n) \] Thus, the asymptotic complexity is: \[ T(n) = \Theta(2^n) \] Final Answer: \[ \boxed{\Theta(2^n)} \]
The characteristic equation is: \[ x^2 - 5x + 6 = 0 \] Factoring gives: \[ (x - 3)(x - 2) = 0 \] Thus, the roots are \( r_1 = 3 \) and \( r_2 = 2 \). Step 2: General solution for the recurrence relation.
The general solution is: \[ T(n) = C_1(3^n) + C_2(2^n) \] Using the initial conditions \( T(0) = 1 \) and \( T(1) = 2 \), we find: \[ C_1 = 0, \quad C_2 \neq 0 \] Step 3: Asymptotic behavior.
Since \( C_1 = 0 \), the solution reduces to: \[ T(n) = C_2(2^n) \] Thus, the asymptotic complexity is: \[ T(n) = \Theta(2^n) \] Final Answer: \[ \boxed{\Theta(2^n)} \]
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