Is $T(1)=1?$

The Gateway to Computer Science Excellence

+1 vote

0

$T\left ( n \right )=2^{n}T\left ( \frac{n}{2} \right )+n^{n}$

$=2^{n}\left ( 2^{n/2}T\left ( \frac{n}{2^{2}} \right )+\left ( \frac{n}{2} \right ) ^{n/2}\right )+n^{n}$

$=2^{n+\frac{n}{2}+\frac{n}{4}+........}T\left ( 1 \right )+n^{n}+\left ( \frac{n}{2} \right )^{n/2}+\left ( \frac{n}{4} \right )^{n/4}+.......1$

$=2^{n}+n^{n}+............$

$=O(n^{n})$

$=2^{n}\left ( 2^{n/2}T\left ( \frac{n}{2^{2}} \right )+\left ( \frac{n}{2} \right ) ^{n/2}\right )+n^{n}$

$=2^{n+\frac{n}{2}+\frac{n}{4}+........}T\left ( 1 \right )+n^{n}+\left ( \frac{n}{2} \right )^{n/2}+\left ( \frac{n}{4} \right )^{n/4}+.......1$

$=2^{n}+n^{n}+............$

$=O(n^{n})$

- All categories
- General Aptitude 1.9k
- Engineering Mathematics 7.5k
- Digital Logic 2.9k
- Programming and DS 4.9k
- Algorithms 4.3k
- Theory of Computation 6.2k
- Compiler Design 2.1k
- Databases 4.1k
- CO and Architecture 3.4k
- Computer Networks 4.1k
- Non GATE 1.5k
- Others 1.5k
- Admissions 595
- Exam Queries 576
- Tier 1 Placement Questions 23
- Job Queries 72
- Projects 17

50,647 questions

56,497 answers

195,490 comments

100,818 users