ITEC 122 2008fall ibarland

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hw07
sums

1. (2pts) Rosen p.160 #2. (What is a₈?)
   1. a_n = 2^n-1
   2. a_n = 7
   3. a_n = (-1)ⁿ+1
   4. a_n = -(-2)ⁿ
2. (10pts) Rosen p.161 #14 (5ed: p.236 #14): sums over S={1,3,5,7}.
   Then, repeat parts (a),(b),(d) summing over S′ = [0,100] ∩ Z. (For (b), refer to Table 2 in the book.) Each answer is a number; you don't need to show your work if you're confident about the answer. No calculator required.
3. (12pts) p.162 #18a-c (5ed p.236 #18a-c) variant: replace "3" with "300" and "2" with "200".
   Leave your answer as arithmetic (i.e. your answer should include terms like “200·301·150”, but it should not include any “…” -- only things you can type into a calculator.)
   You don't need to show your work. (And if you do, clearly circle your answer.)
   Hint: Remember that you can distribute Sigma over addition/subtraction: Σ(f(i)+g(i)) = (Σf(i))+(Σg(i)), regardless of what values i ranges over¹ (Put another way: it doesn't matter which order you do additions in.)

4. In Wii Fit Soccer, you play a goalie; soccer balls are kicked at you, and you get points for every ball you successfully block. (Here's a video trailer.)

   Details:

   • A total of 80 balls are kicked at you (one at a time), over the course of the game. (The upper-left of the screen shows the balls remaining.)
   • If you block multiple balls in a row, your score is higher: You get 1pt for the first ball you block, 2pts for the 2nd in a row you successfully block, etc.. If you miss a ball (ruining your streak), then your next successful block is worth 1pt, then 2pts, etc..
   • The points-per-ball caps out at 10pts. That is, you keep earning 10pts for your 11th, 12th, etc. hit in a row (instead of continuing to rack up 11pts, 12pts, etc..)
   • We'll ignore the penalty objects in the game (cleats, panda-heads).
   1. (1pt) What is the maximum possible score, in this game?
   2. (3pts) Suppose you play the game, and miss exactly one ball. (Way to go!). What is your maximum possible score in this case? Describe how this score would be achieved.
   3. (3pts) Suppose you play the game, and miss exactly one ball. What is your minimum possible score in this case? Describe how this score would be achieved. (Missing which ball would achieve this score?)
   4. (3pts) In a program, we wouldn't want to use magic numbers like 80 and 10; we'd declare named constants name something like N and MAX_PTS_PER_BLOCK, respectively. Give your answers to (a)-(c) above in terms of these terms (Use the names “N” and “m” for these quantities.)

   5. Extra-credit: Prove that these bounds are correct. Hint: Suppose I missed only the ith ball; derive a formula mx(i) for the max number of points achievable (which might have several cases, depending on what i is). Then, show mathematically that in all cases, your answer to part (b) is always ≥ than your formula, for any i. Then do similarly for part (c).

   6. Extra-credit: Implement two functions, maxPossible and minPossible. The description of maxPossible is given here, and minPossible is analogous.

      /** Return the maximum possible Wii Fit Soccer score, * given how many balls were missed. * @param n The number of balls in a game. * @param k The number of balls missed. 0 ≤ k ≤ n. * @param m The maximum number of points possible for a single ball. * @return the maximum possible Wii Fit Soccer score, * if exactly k balls were missed. */ int maxPossible( int numMisses, int N, int m )

      Although I gave those signatures in Java, you can use any programming language of your choice, but you must provide a solid test suite, and your program must pass the test suite!

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©2008, Ian Barland, Radford University Last modified 2008.Dec.13 (Sat)

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