Discussion
Let \(T(n)\) be the recurrence relation defined as follows:
\(T(0) = 1\),
\(T(1) = 2\), and
\(T(n) = 5T(n - 1) - 6T(n - 2)\) for \(n \geq 2\)
Which one of the following statements is TRUE?
Answer: A
The true statement is \(T(n) = \Theta(2^{n})\).
The characteristic equation of \(T(n) = 5T(n-1) - 6T(n-2)\) is \(r^{2} - 5r + 6 = 0\), with roots 2 and 3, so \(T(n) = a2^{n} + b3^{n}\). The initial values fix the constants:
\[ \begin{aligned} T(0) &= a + b = 1 \\ T(1) &= 2a + 3b = 2 \\ \Rightarrow b &= 0, \quad a = 1 \end{aligned} \]
So \(T(n) = 2^{n}\) exactly (check: \(T(2) = 10 - 6 = 4\)). The \(3^{n}\) term would normally dominate, which is the trap, but these initial conditions make its coefficient zero. The \(n2^{n}\) and \(n3^{n}\) forms arise only from a repeated root, which this equation does not have.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development