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If necessary, they need to be done using Maple software or Microsoft Equation 3.0			Document Preview:				Exercise #12)  Show that [x]+ [x + 1/2]= [2x] whenever x is a real number.    Exercise #22)  Conjecture a formula for the nth term of {an} if the first ten terms of this sequence are as follows:  a) 3, 11, 19, 27, 35, 43, 51, 59, 67, 75  b) 5, 7, 11, 19, 35, 67, 131, 259, 515, 1027  c) 1, 0, 0, 1, 0, 0, 0, 0, 1, 0 d) 1, 3, 4, 7, 11, 18, 29, 47, 76, 123  Exercise #6)  By putting together two triangular arrays, one with n rows and one with n - 1 rows, to form a square (as illustrated for n = 4), show that tn-1 + tn = n2, where tn is the nth triangular number. (it’s a 4 by 4 square with 16 dots in it)- fyi  Exercise #10)  Show that p1=1and pk=pk-1+(3k -2) for k =2.   Conclude that                                   pn= knk=1 (3k -2) and evaluate this sum to find a simple formula for pn.    Exercise #18)  Find n! for n equal to each of the first ten positive integers.  Exercise #8)  Use mathematical induction to prove that j=1n j3 = 13+ 23 + 33. . . + n3 = [n(n + 1)/2]² for every positive integer n. Exercise # 14)  Show that any amount of postage that is an integer number of cents greater than 53 cents can be formed using just 7-cent and 10-cent stamps.  Exercise # 20)  Use mathematical induction to prove that 2n 	Exercise #12)  Show that [x]+ [x + 1/2]= [2x] whenever x is a real number.   Exercise #22)  Conjecture a formula for the nth term of {an} if the first ten terms of this sequence are as follows:  a) 3, 11, 19, 27, 35, 43, 51, 59, 67, 75  b) 5, 7, 11, 19, 35, 67, 131, 259, 515, 1027  c) 1, 0, 0, 1, 0, 0, 0, 0, 1, 0 d) 1, 3, 4, 7, 11, 18, 29, 47, 76, 123 Exercise #6)  By putting together two triangular arrays, one with n rows and one with n − 1 rows, to form a square (as illustrated for n = 4), show that tn−1 + tn = n2, where tn is the nth triangular number. (it’s a 4 by 4 square with 16 dots in it)- fyi Exercise #10)  Show that p1=1and pk=pk−1+(3k −2) for k ≥2.   Conclude that                                   pn=  (3k −2) and evaluate this sum to find a simple formula for pn.   Exercise #18)  Find n! for n equal to each of the first ten positive integers. Exercise #8)  Use mathematical induction to prove that  = +  + . . . +  = [n(n + 1)/2]² for every positive integer n. Exercise # 14)  Show that any amount of postage that is an integer number of cents greater than 53 cents can be formed using just 7-cent and 10-cent stamps. Exercise # 20)  Use mathematical induction to prove that   1. Find such representation for as many positive integers n as you can. Exercise 1.4 I)  Find the Fibonacci numbers f100, f200, and f500. Exercise 1.4 J)  A surprising theorem states that the Fibonacci numbers are the positive values of the polynomial 2x +  − 2 −  −  y + 2y as x and y range over all nonnegative integers. Verify this conjecture for the values of x and y where x and y are nonnegative integers with x + y ≤ 100. Exercise 1.5 K)  Verify the Collatz conjecture described in the preamble to Exercise 49 (Find the sequence obtained by iterating T starting with n = 39)for all integers n not exceeding 10,000. SECTION 2 Exercise #2)  Convert (89156)10 from decimal to base 8 notation. Convert (706113)8 from base 8 to decimal notation. Exercise #4)  Convert (101001000)2 from binary to decimal notation and (1984)10 from decimal to binary notation. Exercise #2)  Add (10001000111101)2 and (11111101011111)2. Exercise #4)  Subtract (101110101)2 from (1101101100)2. Exercise #6)  Multiply (1110111)2 and (10011011)2. Exercise #8)  Find the quotient and remainder when (110100111)2 is divided by (11101)2. Exercise A)  Find the binary, octal, and hexadecimal expansions of each of the following integers. a) 9876543210  b) 1111111111  c) 10000000001 Exercise B)  Find the decimal expansion of each of the following integers. a) (1010101010101)2  b) (765432101234567)8 c) (ABBAFADACABA)16 Exercise C)  Evaluate each of the following sums, expressing your answer in the same base used to represent the summands. a) (11011011011011011)2 + (1001001001001001001001)2 b) (12345670123456)8 + (765432107654321)8 c) (123456789ABCD)16 + (BABACACADADA)16 1

Robert · Accepted Answer

1 

Exercise 1.3 E)   
Solution: For n = 1, the statement reduces to 1
2
 = 
6
321 
and is obviously true. 
Assuming the statement is true for n = k: 
1
2
 + 2
2
 + 3
2
 + . . . + k
2
 = 
6
)2)(1(  kkk
 
We will prove that the statement must be true for n = k + 1: 
1
2
 + 2
2
 + 3
2
 + . . . + (k + 1)
2
 = 
6
)32)(2)(1(  kkk
 
The left-hand side of (2) can be written as 
1
2
 + 2
2
 + 3
2
 + . . . + k
2
 + (k + 1)
2
 
In view of (1), this simplifies to: 
(1
2
 + 2
2
 + 3
2
 + . . .

If necessary, they need to be done using Maple software or Microsoft Equation 3.0 Document Preview: Exercise #12) Show that [x]+ [x + 1/2]= [2x] whenever x is a real number. Exercise #22) Conjecture a...

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