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Questions and Answers of Statistics

Showthat if a1, a2, . . . , an are n distinct real numbers, exactly n − 1 multiplications are used to compute the product of these n numbers no matter how parentheses are inserted into their
Let a be an integer and d be a positive integer. Show that the integers q and r with a = dq + r and 0 ≤ r < d, which were shown to exist in Example 5, are unique.
Can you use the well-ordering property to prove the statement: "Every positive integer can be described using no more than fifteen English words"? Assume the words come from a particular dictionary
Show that the well-ordering property can be proved when the principle of mathematical induction is taken as an axiom.
Show that we can prove the well-ordering property when we take strong induction as an axiom instead of taking the well-ordering property as an axiom.
a) Determine which amounts of postage can be formed using just 4-cent and 11-cent stamps. b) Prove your answer to (a) using the principle of mathematical induction. Be sure to state explicitly your
Which amounts of money can be formed using just twodollar bills and five-dollar bills? Prove your answer using strong induction.
Use strong induction to prove that √2 is irrational.
Give a recursive definition of Pm(n), the product of the integer m and the nonnegative integer n.
Give a recursive definition of the set of positive integers that are multiples of 5.
Give a recursive definition of a) The set of even integers. b) The set of positive integers congruent to 2 modulo 3. c) The set of positive integers not divisible by 5.
Let S be the subset of the set of ordered pairs of integers defined recursively by Basis step: (0, 0) ∈ S. Recursive step: If (a, b) ∈ S, then (a, b + 1) ∈ S, (a + 1, b + 1) ∈ S, and (a + 2,
Define well-formed formulae of sets, variables representing sets, and operators from {-, ∪, ∩, −}.
Give a recursive definition of the reversal of a string.
Give a recursive definition of wi, where w is a string and i is a nonnegative integer. (Here wi represents the concatenation of i copies of the string w.)
When does a string belong to the set A of bit strings defined recursively by λ ∈ A 0x1 ∈ A if x ∈ A, where λ is the empty string?
Use Exercise 37 and mathematical induction to show that l(wi) = i · l(w), where w is a string and i is a nonnegative integer.
Use structural induction to show that n(T) ≥ 2h(T) + 1, where T is a full binary tree, n(T) equals the number of vertices of T, and h(T) is the height of T.
Involve this version of Ackermann's function. Show that A(m, 2) = 4 whenever m ≥ 1.
Use strong induction to prove that a function F defined by specifying F(0) and a rule for obtaining F(n + 1) from the values F(k) for k = 0, 1, 2, . . . , n is well defined.
Give a recursive definition of the sequence {an}, n = 1, 2, 3, . . . if a) an = 6n. b) an = 2n + 1. c) an = 10n. d) an = 5.
Let F be the function such that F(n) is the sum of the first n positive integers. Give a recursive definition of F(n).
Trace Algorithm 1 when it is given n = 5 as input. That is, show all steps used by Algorithm 1 to find 5!, as is done in Example 1 to find 4!.
Give a recursive algorithm for finding the minimum of a finite set of integers, making use of the fact that the minimum of n integers is the smaller of the last integer in the list and the minimum of
Give a recursive algorithm for finding n! modm whenever n and m are positive integers.
Devise a recursive algorithm for computing the greatest common divisor of two nonnegative integers a and b with a < b using the fact that gcd(a, b) = gcd(a, b − a).
Describe a recursive algorithm for multiplying two nonnegative integers x and y based on the fact that xy = 2(x · (y/2)) when y is even and xy = 2(x · [y/2]) + x when y is odd, together with the
Prove that Algorithm 3 for computing gcd(a, b) when a and b are positive integers with a < b is correct.
Prove that the recursive algorithm that you found in Exercise 7 is correct.
Devise a recursive algorithm for computing n2 where n is a nonnegative integer, using the fact that (n + 1)2 = n2 + 2n + 1. Then prove that this algorithm is correct.
How does the number of multiplications used by the algorithm in Exercise 24 compare to the number of multiplications used by Algorithm 2 to evaluate a2n?
How does the number of multiplications used by the algorithm in Exercise 26 compare to the number of multiplications used by Algorithm 2 to evaluate an?
Devise a recursive algorithm to find the nth term of the sequence defined by a0 = 1, a1 = 2, and an = an−1 · an−2, for n = 2, 3, 4, . . . .
Trace Algorithm 3 when it finds gcd(8, 13). That is, show all the steps used by Algorithm 3 to find gcd(8, 13).
Is the recursive or the iterative algorithm for finding the sequence in Exercise 29 more efficient?
Devise an iterative algorithm to find the nth term of the sequence defined in Exercise 32.
Give iterative and recursive algorithms for finding the nth term of the sequence defined by a0 = 1, a1 = 3, a2 = 5, and an = an−1 · a2n−2 · a3n−3. Which is more efficient?
Give a recursive algorithm for finding the reversal of a bit string. (See the definition of the reversal of a bit string in the preamble of Exercise 34 in Section 5.3.)
Prove that the recursive algorithm for finding the reversal of a bit string that you gave in Exercise 37 is correct.
Give a recursive algorithm for tiling a 2n × 2n checkerboard with one square missing using right triominoes.
Give a recursive algorithm for computing values of the Ackermann function.
Use a merge sort to sort b, d, a, f, g, h, z, p, o, k into alphabetic order. Show all the steps used by the algorithm.
Show that for all positive integers m and n there are sorted lists with m elements and n elements, respectively, such that Algorithm 10 uses m + n − 1 comparisons to merge them into one sorted list.
Prove that the merge sort algorithm is correct. The quick sort is an efficient algorithm. To sort a1, a2, . . . , an, this algorithm begins by taking the first element a1 and forming two sublists,
Trace Algorithm 4 when it is given m = 5, n = 11, and b = 3 as input. That is, show all the steps Algorithm 4 uses to find 311mod 5.
Let a1, a2, . . . , an be a list of n distinct real numbers. How many comparisons are needed to form two sublists from this list, the first containing elements less than a1 and the second containing
What is the largest number of comparisons needed to order a list of four elements using the quick sort algorithm?
Determine the worst-case complexity of the quick sort algorithm in terms of the number of comparisons used.
Give a recursive algorithm for computing nx whenever n is a positive integer and x is an integer, using just addition.
Give a recursive algorithm for finding the sum of the first n odd positive integers.
Prove that the program segment y: = 1 z: = x + y is correct with respect to the initial assertion x = 0 and the final assertion z = 1.
Suppose that both the program assertion p{S}q0 and the conditional statement q0 → q1 are true. Show that p{S}q1 also must be true.
Use a loop invariant to verify that the Euclidean algorithm (Algorithm 1 in Section 4.3) is partially correct with respect to the initial assertion "a and b are positive integers" and the final
Verify that the program segment x := 2 z := x + y if y > 0 then z := z + 1 else z := 0 is correct with respect to the initial assertion y = 3 and the final assertion z = 6.
a) Can you use the principle of mathematical induction to find a formula for the sum of the first n terms of a sequence? b) Can you use the principle of mathematical induction to determine whether a
a) What is a recursive algorithm? b) Describe a recursive algorithm for computing the sum of n numbers in a sequence.
Use the merge sort algorithm to put the list 4, 10, 1, 5, 3, 8, 7, 2, 6, 9 in increasing order.
a) Does testing a computer program to see whether it produces the correct output for certain input values verify that the program always produces the correct output? b) Does showing that a computer
a) For which positive integers n is 11n + 17 ≤ 2n? b) Prove the conjecture you made in part (a) using mathematical induction.
a) Which amounts of postage can be formed using only 5-cent and 9-cent stamps? b) Prove the conjecture you made using mathematical induction. c) Prove the conjecture you made using strong
a) State the well-ordering property for the set of positive integers. b) Use this property to show that every positive integer greater than one can be written as the product of primes.
Use mathematical induction to show that 2/3 + 2/9 + 2/27 + ∙ ∙ ∙ + 2/3n = 1 - 1/3n whenever n is a positive integer.
Use mathematical induction to prove that 43 divides 6n+1 + 72n−1 for every positive integer n.
Use mathematical induction to prove that 64 divides 32n+2 + 56n + 55 for every positive integer n.
Show that if n is a positive integer, then
Suppose that f (x) = ex and g(x) = xex. Use mathematical induction together with the product rule and the fact that f'(x) = ex to prove that g(n)(x) = (x + n)ex whenever n is a positive integer.
Formulate a conjecture about which Fibonacci numbers are even, and use a form of mathematical induction to prove your conjecture.
Prove that fkfn + fk+1fn+1 = fn+k+1 for all nonnegative integers n and k, where f1 denotes the ith Fibonacci number.
Show that l20 + l21 +· · ·+ l2n = lnln+1 + 2 whenever n is a nonnegative integer and l1 is the ith Lucas number.
Use mathematical induction to show that (cos x + i sin x)n = cos nx + i sin nx whenever n is a positive integer. (Here i is the square root of −1.)
Use mathematical induction to prove that Σnj=1 j22j = n22n+1 − n2n+2 + 3 · 2n+1 − 6 for every positive integer n.
Show if n is a positive integer with n ‰¥ 2, then
Use mathematical induction to show that 1 · 20 + 2 · 21 + 3 · 22 +· · ·+n · 2n−1 = (n − 1) · 2n + 1 whenever n is a positive integer.
A lattice point in the plane is a point (x, y) where both x and y are integers. Use mathematical induction to show that at least n + 1 straight lines are needed to ensure that every lattice point (x,
Suppose that B = MAM−1, where A and B are n × n matrices and M is invertible. Show that Bk = MAkM−1 for all positive integers k. (Consult both the text of Section 2.6 and the preamble to
Show that n! can be represented as the sum of n of its distinct positive divisors whenever n ≥ 3.
Use mathematical induction to prove that if n people stand in a line, where n is a positive integer, and if the first person in the line is a woman and the last person in line is a man, then
Use mathematical induction to show that when n circles divide the plane into regions, these regions can be colored with two different colors such that no regions with a common boundary are colored
Show that if n is a positive integer, then
Use mathematical induction to show that if n is a positive integers, the sequence 2 mod n, 22 mod n, 222 mod n, 2222 modn, . . . is eventually constant (that is, all terms after a finite number of
By successively using the defining rule for M(n), find a) M(102). b) M(101). c) M(99). d) M(97). e) M(87). f) M(76).
Is this proof thatwhenever n is a positive integer, correct? Justify your answer. Basis step: The result is true when n = 1 because Inductive step: Assume that the result is true for n. Then Hence,
Show that n circles divide the plane into n2 − n + 2 regions if every two circles intersect in exactly two points and no three circles contain a common point.
Use the well-ordering property to show that √2 is irrational.
a) Show that if a1, a2, . . . , an are positive integers, then gcd(a1, a2, . . . , an−1, an) = gcd(a1, a2, . . . , an−2, gcd(an−1, an)). b) Use part (a), together with the Euclidean algorithm,
Find an explicit formula for f (n) if f (1) = 1 and f (n) = f (n − 1) + 2n − 1 for n ≥ 2. Prove your result using mathematical induction.
Let S be the set of bit strings defined recursively by λ ∈ S and 0x ∈ S, x1 ∈ S if x ∈ S, where λ is the empty string. a) Find all strings in S of length not exceeding five. b) Give an
Show that (()()) is a balanced string of parentheses and (())) is not a balanced string of parentheses.
Find all balanced strings of parentheses with four or fewer symbols.
Finda) N(()).b) N( )))())(().c) N((()(()).d) N()((()))(())).
Give a recursive algorithm for finding all balanced strings of parentheses containing n or fewer symbols.
Verify the program segment if x > y then x := y with respect to the initial assertion T and the final assertion x ≤ y.
Devise a recursive algorithm that counts the number of times the integer 0 occurs in a list of integers.
Use mathematical induction to show that 2n > n3 whenever n is an integer greater than 9.
Prove that this sequence is well defined. That is, show that a(n) is uniquely defined for all nonnegative integers n. Deal with some unusual sequences, informally called self-generating sequences,
Use the formula from Exercise 72 to show that a(n) = a(n − 1) if μn - [μn] < 1 − μ and a(n) = a(n − 1) + 1 otherwise.
Find the first 10 terms of both the sequences m(n) and f (n) defined by the following pair of interwoven recurrence relations: m(n) = n − f (m(n − 1)), f (n) = n − m(f (n − 1)) for n ≥ 1, f
Use mathematical induction to prove that a − b is a factor of an − bn whenever n is a positive integer.
There are 18 mathematics majors and 325 computer science majors at a college. a) In how many ways can two representatives be picked so that one is a mathematics major and the other is a computer
How many bit strings of length ten both begin and end with a 1?

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