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# lec0214 - CS 173 Discrete Mathematical Structures Cinda...

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CS 173: Discrete Mathematical Structures Cinda Heeren [email protected] Rm 2213 Siebel Center Office Hours: W 9:30-11:30a

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Cs173 - Spring 2004 CS 173 Announcements Homework 5 available.  Due 02/19, 8a. Midterm 1:  2/23/06, 7-9p, location SC 1404. Send me an email asap if you have a conflict. Sections next week will be exam review.   Three additional reviews:   Tues, 2/21, 6-7p, SC 1129 Thur, 2/23, 9:30-10:45a, SC 1404 Thur, 2/23, 11:00a-12:15, SC 1214
Cs173 - Spring 2004 CS 173 Summation How do you know this is true? ca i + b i ( ) i =1 k = c a i i =1 k + b i i =1 k Use associativity to separate the bs from the as. Use distributivity to factor the cs.

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Cs173 - Spring 2004 CS 173 Summations you should know… What is S = 1 + 2 + 3 + … + n? You get n copies of (n+1).  But we’ve over added by a factor of 2!   So just divide by 2. S = 1 + 2 + + n S = n + n-1 + + 1 2s = n+1 + n+1 + + n+1 Write the sum. Write it again. Add together. k k =1 n = n ( n +1) 2
Cs173 - Spring 2004 CS 173 Summations you should know… What is S = 1 + 3 + 5 + … + (2n - 1)? Sum of first n odds. (2 k - 1) k =1 n = 2 k k =1 n - 1 k =1 n = 2 n ( n +1) 2 - n = n 2

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Cs173 - Spring 2004 CS 173 Summations you should know… What is S = 1 + 3 + 5 + … + (2n - 1)? Sum of first n odds. = n 2
Cs173 - Spring 2004 CS 173 Summations you should know… What is S = 1 + r + r 2  + … + r n Geometric Series r k k = 0 n =1+ r +K + r n r r k k = 0 n = r + r 2 +K + r n +1 Multiply by r Subtract 2 nd  from 1 st r k k = 0 n - r r k k = 0 n =1- r n +1 factor (1- r ) r k k = 0 n =1- r n +1 divide r k k = 0 n = 1- r n +1 (1- r ) DONE!

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Cs173 - Spring 2004 CS 173 Summations you should know… What about: r k k = 0 =1+ r +K + r n +K = n lim 1- r n +1 (1- r ) If r
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lec0214 - CS 173 Discrete Mathematical Structures Cinda...

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