let n be a positive integer and let x1,...,xn be real numbers. Prove that ( x1+...+xn)2...
let n be a positive integer and let
x1,...,xn be real numbers. Prove that (
x1+...+xn)2
n(x12+ x22 +...+
xn2).
Homework Answers
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let n be a positive integer and let
x1,...,xn be real numbers. Prove that (
x1+...+xn)2...
5. Let {xn} and {yn} be sequences of real numbers such that x1 = 2 and...
5. Let {xn} and {yn} be sequences of real numbers such that x1 =
2 and y1 = 8 and for n = 1,2,3,···
x2nyn + xnyn2 x2n + yn2 xn+1 = x2 + y2 and yn+1 = x + y .
nn nn
(a) Prove that xn+1 − yn+1 = −(x3n − yn3 )(xn − yn) for all
positive integers n.
(xn +yn)(x2n +yn2) (b) Show that 0 < xn ≤ yn for all positive
integers n.
Hence, prove...
#2. Let n E N and x1,x2,.., Xn, yı,y2,..,Ja, and zł,Zy, #a) Prove the identity An be real numbers #b) Use the identity in #a) to prove (the Cauchy-Schwartz inequality) that #1) Extend the result i...
#2. Let n E N and x1,x2,.., Xn, yı,y2,..,Ja, and zł,Zy, #a) Prove the identity An be real numbers #b) Use the identity in #a) to prove (the Cauchy-Schwartz inequality) that #1) Extend the result in #b) to prove that 4 #d) Use the inequality in #b) to prove the inequality which is the triangle inequality
#2. Let n E N and x1,x2,.., Xn, yı,y2,..,Ja, and zł,Zy, #a) Prove the identity An be real numbers #b) Use the identity in...
3. Let {x1, x2,...,xn} be a list of numbers and let ¯ x denote the average...
3. Let {x1, x2,...,xn} be a list of numbers and let ¯ x denote the average of the list. Let a and b be two constants, and for each i such that 1 ≤ i ≤ n, let yi = axi + b. Consider the new list {y1,y2,...,yn}, and let the average of this list be ¯ y. Prove a formula for ¯ y in terms of a, b, and ¯ x. 4. Let n be a positive integer. Consider...
1. Let Xn ER be a sequence of real numbers. (a) Prove that if Xn is...
1. Let Xn ER be a sequence of real numbers. (a) Prove that if Xn is an increasing sequence bounded above, that is, if for all n, xn < Xn+1 and there exists M E R such that for all n E N, Xn < M, then limny Xn = sup{Xnin EN}. (b) Prove that if Xn is a decreasing sequence bounded below, that is, if for all n, Xn+1 < xn and there exists M ER such that for...
1. (Integers: primes, divisibility, parity.) (a) Let n be a positive integer. Prove that two numbers...
1. (Integers: primes, divisibility, parity.) (a) Let n be a positive integer. Prove that two numbers na +3n+6 and n2 + 2n +7 cannot be prime at the same time. (b) Find 15261527863698656776712345678%5 without using a calculator. (c) Let a be an integer number. Suppose a%2 = 1. Find all possible values of (4a +1)%6. 2. (Integers: %, =) (a) Suppose a, b, n are integer numbers and n > 0. Prove that (a+b)%n = (a%n +B%n)%n. (b) Let a,...
#2. Let n E N and X1,X2, ,yn, and zi,22, An be real numbers. ,An, yī,Y2, #a) Prove the identity #b) Use the identity in #a) to prove (the Cauchy-Schwartz inequality) that #1) Extend the result in...
#2. Let n E N and X1,X2, ,yn, and zi,22, An be real numbers. ,An, yī,Y2, #a) Prove the identity #b) Use the identity in #a) to prove (the Cauchy-Schwartz inequality) that #1) Extend the result in #b) to prove that #d) Use the inequality in #b) to prove the inequality which is the triangle inequality
#2. Let n E N and X1,X2, ,yn, and zi,22, An be real numbers. ,An, yī,Y2, #a) Prove the identity #b) Use the identity...
Let n be a positive integer and let F = {X 5 [n]: X|2|[n] \X]} Prove...
Let n be a positive integer and let F = {X 5 [n]: X|2|[n] \X]} Prove that F is a maximum intersecting family.
2. a. Letỉ be the median of X1, , xn, n odd. Prove that the identity...
2. a. Letỉ be the median of X1, , xn, n odd. Prove that the identity 1-1 1-1 Hold if and only if z b. Let X1, , Xn be a random sample form f(p, b), where f(p, b) is the Laplace distribution with density 1 2h2 -k-시 Assumingthat b is known and that n is odd, show that the MLE of μ is the sample median, X. (Hint: Use (a).)
Let n be a positive integer and a,b,c be real numbers greater than 1. Select all...
Let n be a positive integer and a,b,c be real numbers greater than 1. Select all of the statements that are true. In(a) In(b) = In for all ab > 1. (In(a)) = bin(a) In(a) = £In(a) In(a? +62)= 2ln(a) + 2in(b) In() - = In(a) - In(b) – In (c) s In(a-")= -nIn(a)
Let (xn) be a bounded sequence of real numbers, and put u = lim supn→∞ xn . Let E be the set consisting of the limits of all convergent subsequences of (xn). Show that u ∈ E and that u = sup(E). Form...
Let (xn) be a bounded sequence
of real numbers, and put u = lim supn→∞ xn . Let E be the set
consisting of the limits of all convergent subsequences of (xn).
Show that u ∈ E and that u = sup(E).
Formulate and prove a similar result for lim infn→∞ xn .
Thank you!
7. Let (Fm) be a bounded sequence of real numbers, and put u-lim supn→oorn . Let E be the set consisting of the limits of...
5. Let {xn} and {yn} be sequences of real numbers such that x1 =
2 and y1 = 8 and for n = 1,2,3,···
x2nyn + xnyn2 x2n + yn2 xn+1 = x2 + y2 and yn+1 = x + y .
nn nn
(a) Prove that xn+1 − yn+1 = −(x3n − yn3 )(xn − yn) for all
positive integers n.
(xn +yn)(x2n +yn2) (b) Show that 0 < xn ≤ yn for all positive
integers n.
Hence, prove...
#2. Let n E N and x1,x2,.., Xn, yı,y2,..,Ja, and zł,Zy, #a) Prove the identity An be real numbers #b) Use the identity in #a) to prove (the Cauchy-Schwartz inequality) that #1) Extend the result in #b) to prove that 4 #d) Use the inequality in #b) to prove the inequality which is the triangle inequality
#2. Let n E N and x1,x2,.., Xn, yı,y2,..,Ja, and zł,Zy, #a) Prove the identity An be real numbers #b) Use the identity in...
1. Let Xn ER be a sequence of real numbers. (a) Prove that if Xn is an increasing sequence bounded above, that is, if for all n, xn < Xn+1 and there exists M E R such that for all n E N, Xn < M, then limny Xn = sup{Xnin EN}. (b) Prove that if Xn is a decreasing sequence bounded below, that is, if for all n, Xn+1 < xn and there exists M ER such that for...
1. (Integers: primes, divisibility, parity.) (a) Let n be a positive integer. Prove that two numbers na +3n+6 and n2 + 2n +7 cannot be prime at the same time. (b) Find 15261527863698656776712345678%5 without using a calculator. (c) Let a be an integer number. Suppose a%2 = 1. Find all possible values of (4a +1)%6. 2. (Integers: %, =) (a) Suppose a, b, n are integer numbers and n > 0. Prove that (a+b)%n = (a%n +B%n)%n. (b) Let a,...
#2. Let n E N and X1,X2, ,yn, and zi,22, An be real numbers. ,An, yī,Y2, #a) Prove the identity #b) Use the identity in #a) to prove (the Cauchy-Schwartz inequality) that #1) Extend the result in #b) to prove that #d) Use the inequality in #b) to prove the inequality which is the triangle inequality
#2. Let n E N and X1,X2, ,yn, and zi,22, An be real numbers. ,An, yī,Y2, #a) Prove the identity #b) Use the identity...
Let n be a positive integer and let F = {X 5 [n]: X|2|[n] \X]} Prove that F is a maximum intersecting family.
2. a. Letỉ be the median of X1, , xn, n odd. Prove that the identity 1-1 1-1 Hold if and only if z b. Let X1, , Xn be a random sample form f(p, b), where f(p, b) is the Laplace distribution with density 1 2h2 -k-시 Assumingthat b is known and that n is odd, show that the MLE of μ is the sample median, X. (Hint: Use (a).)
Let n be a positive integer and a,b,c be real numbers greater than 1. Select all of the statements that are true. In(a) In(b) = In for all ab > 1. (In(a)) = bin(a) In(a) = £In(a) In(a? +62)= 2ln(a) + 2in(b) In() - = In(a) - In(b) – In (c) s In(a-")= -nIn(a)
Let (xn) be a bounded sequence
of real numbers, and put u = lim supn→∞ xn . Let E be the set
consisting of the limits of all convergent subsequences of (xn).
Show that u ∈ E and that u = sup(E).
Formulate and prove a similar result for lim infn→∞ xn .
Thank you!
7. Let (Fm) be a bounded sequence of real numbers, and put u-lim supn→oorn . Let E be the set consisting of the limits of...
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