q7-sequences - not equal to any one of r 1 , r 2 , r 3 ,...

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CSE 260 QUIZ-7– Sequences-summations: ANSWERS (25 minutes) NAME: 1. What are the nth terms of the following sequences. (a) 1,3,5,7,9,. . . 2 n - 1 (b) 2,4,6,8,10,. . . 2 n (c) 3,6,12,24,48,96,192,. . . 3 . 2 n - 1 (d) 1,0,2,0,4,0,8,0,16,0,. . . 2 ( n - 1) / 2 for odd terms 0 for even terms 2. Compute the following double sum. Σ 3 i =1 Σ 2 j =1 ( i - j ) 3 i =1 (( i - 1) + ( i - 2)) =(1 - 1) + (1 - 2) + (2 - 1) + (2 - 2) + (3 - 1) + (3 - 2) =3 3. Find the value of Σ 200 k =100 k Given Σ n k =1 k = n ( n + 1) / 2 Σ 200 k =1 k - Σ 99 k =1 k =(200(200 + 1)) / 2 - (99(99 + 1)) / 2 =20100 - 4950 =15150 1
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4. In showing that the set of all real numbers is uncountable , we show that the subset of all real numbers that fall between 0 and 1 is also uncountable . We show this by contradiction assuming that the subset is countable . Under this assumption, the real numbers between 0 and 1 can be listed in some order, say r 1 , r 2 , r 3 , .... Let the decimal representation of these numbers be r 1 = 0 .d 11 d 12 d 13 d 14 ... r 2 = 0 .d 21 d 22 d 23 d 24 ... r 3 = 0 .d 31 d 32 d 33 d 34 ... - - - where d ij ² { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } We contradict by deFning a real number r that is between 0 and 1 but
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Unformatted text preview: not equal to any one of r 1 , r 2 , r 3 , .... (a) Complete the following deFnition of r: d i = 4 if d ii 6 = 4 d i = 5 if d ii = ?4 (b) Indicate why r is not equal to any of r 1 , r 2 , r 3 , ..... ith digit of r is dierent from the ith digit of r i i 5. Indicate if the following sets are countable . (a) Set of all odd positive integers. Yes, countable because of the one to one correspondence between Z + and the set of all odd positive integers, as shown below. 1 2 3 4 . . . ^ ^ ^ ^ | | | | v v v v 1 3 5 7. . . (b) Set of all odd integers. 2 Yes, countable because of the one to one correspondence between Z + and the set of all odd positive integers, as shown below. 1 2 3 4 5 6. . . ^ ^ ^ ^ ^ ^ | | | | | | v v v v v v 1-1 2-2 3-3 . . . 3...
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This note was uploaded on 06/08/2009 for the course CSE 260 taught by Professor Saktipramanik during the Spring '08 term at Michigan State University.

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q7-sequences - not equal to any one of r 1 , r 2 , r 3 ,...

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