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1. (‡) Prove asymptotic bounds for the following recursion relations. Tighter bounds will receive more marks. You may use the Master Theorem if it applies.
1. C(n) = 3C(n/2) + n
2. G(n) = G(n - 1) + 1/n
3. I(n) = I(n/2) + n/ lg(n)
2. Define a (p,q)-tree as a rooted tree where every internal node has between p and q (inclusive) children. Use the Master Theorem to give asymptotic bounds for the height of the tree. You can assume both p and q are constants with 2 ≤ p ≤ q.
3. (‡) Dominos
A 2 × 10 rectangle filled with ten dominos, and a 2 × 2 × 10 box filled with ten slabs.
1. A domino is a 2×1 or 1×2 rectangle. How many different ways are there to completely fill a 2 × n rectangle with n dominos?
2. A slab is a three-dimensional box with dimensions 1 × 2 × 2, 2 × 1 × 2, or 2 × 2 × 1. How many different ways are there to fill a 2 × 2 × n box with n slabs? Set up a recurrence relation and give reasonable exponential upper and lower bounds.
The quotient of 3d 3 and 9d 5 is The key word quotient means division so the problem becomes 1d 3 -5/ 5. Divide the coef?cients: 1d 3 /3d-5 . While dividing like bases, subt
Erin is painting a bathroom along with four walls each measuring 8 ft through 5.5 ft. Ignoring the doors or windows, what is the area to be painted? The area of the room is the
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y=2x+3=
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Prove that in any triangle the sum of the squares of any two sides is equal to twice the square of half of the third side together with twice the square of the median, which bisect
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