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Course Queries Syllabus Queries 2 years ago
Posted on 16 Aug 2022, this text provides information on Syllabus Queries related to Course Queries. Please note that while accuracy is prioritized, the data presented might not be entirely correct or up-to-date. This information is offered for general knowledge and informational purposes only, and should not be considered as a substitute for professional advice.
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In a syllabus of mine, they try to find a closed form of the following recurrence relation
The method I usually use to find the closed form of a recurrence is expand it a few times and try to find a pattern. Then I verify that pattern using induction.
In my syllabus they only show the verification part, by using T(2l)≤c(3l−2l)T(2l)≤c(3l−2l) as hypothesis, where cc is chosen such T(2)≤cT(2)≤c and l≥1l≥1.
So, I tried expanding the recurrence relation to see where I could find that pattern, but I don't get anything close to it.
For example:
Hint: Consider S(k)=T(2k)+c⋅2kS(k)=T(2k)+c⋅2k. Show that S(k+1)⩽3S(k)S(k+1)⩽3S(k). Compute S(0)S(0). Deduce that S(k)⩽3k(1+c)S(k)⩽3k(1+c) for every k⩾0k⩾0 and, finally, that T(2k)⩽3k+c⋅(3k−2k)T(2k)⩽3k+c⋅(3k−2k) for every k⩾0k⩾0.
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