prove that |a|^2 = a^2 for all a ∈ R be sure to do both positive...
prove that |a|^2 = a^2 for all a ∈ R
be sure to do both positive and negative exponents
this question from analysis 1
Homework Answers
Do both please will thumb up Prove that n2 +1 > 2 for any positive integer...
Do both please will thumb up
Prove that n2 +1 > 2 for any positive integer n < 4. Use induction to prove: > 1.22 = (n-1)20+1 + 2,Vn e Z,n 1
1. Recall that x E R is positive (resp. negative) if x = (an) which is...
1. Recall that x E R is positive (resp. negative) if x = (an) which is positively (resp textitnegatively) bounded away from 0. Prove the following LIM00 an for a Cauchy sequence n-oo (a) For any E R, exactly one of the following is true: x is positive, is negative, or x= 0 E R is positive if and only if -x is negative. (b) (c) If x, y E R are both positive, then x + y and xy...
Prove the following version of the division algorithm, which holds for both positive and negative divisors....
Prove the following version of the division algorithm, which holds for both positive and negative divisors. Extended Division Algorithm: Let a and b be integers with b = 0. Then there exist unique integers q and r such that a = bq + r and 0 sr<|bl| [ Hint: Apply Theorem 1.1 when a divided by [b]. Then consider two cases (b >0 and b < 0) Explain the answer and visible for read
Bonus question: 4 bonus marks] A positive integer r is called powerful if for all prime numbers P...
please post clear picture or solution.
Bonus question: 4 bonus marks] A positive integer r is called powerful if for all prime numbers P, p implies p | r. A positive integer z is called a perfect power if there exist a prime number p and a natural number n such that p". An Achilles number is one that is powerful but is not a perfect power. For example, 72 is an Achilles number. Prove that if a and b...
PLEASE DO BOTH (5) (5 pts) Prove the identity (1) () = (x) (---), whenever n,...
PLEASE DO BOTH
(5) (5 pts) Prove the identity (1) () = (x) (---), whenever n, r, and k are non-negative integers with rn andk Sr. (6) (5 pts) How many ways are there to select five unordered elements from a set with three elements when repetition is allowed?
7. Prove that for any positive real number r, if r is not an integer, then...
7. Prove that for any positive real number r, if r is not an integer, then [x]+-1= 1
answer all parts please! :) 1. Recall that (an) which is positively (resp textitnegatively) bounded away...
answer all parts please! :)
1. Recall that (an) which is positively (resp textitnegatively) bounded away from 0. Prove the following: eR is positive resp. negative) if x = LIMn0 an for a Cauchy sequence R, exactly one of the following is true (a) For any x is positive, r is negative, or (b) xRis positive if and only if -x is negative also positive (c) If ar, y E R are both positive, then r + y and ry...
k is a constant, it can be positive or negative 3. Prove that the quantity s--k...
k is a constant, it can be
positive or negative
3. Prove that the quantity s--k Σ P, In(R) r=1 is a maximum when P, = 1/n for all r. Do this by showing that S-kinn-0. Hint: The inequality In [1/(nP )] 1/(nP,) 1 will be helpful. TO arrive at an expression in which you can use this inequality, you will have to insert Ση_B-1 in an appropriate place. This is a problem which is easier not using calculus because...
prove that for all natural numbers r, chi squared ( 2 r ) = t (...
prove that for all natural numbers r, chi squared ( 2 r ) = t ( r, 1/2 )
Please do both questions. wrong answers will be given thumbs down. Question 7. Prove using the...
Please do both questions. wrong answers will be given thumbs
down.
Question 7. Prove using the Division Lemma that Yn E Z, n3 n is divisible by 3 (any proof not using the Division Lemma will receive no credit). Question 8. Define a relation ~ on R \ {0} by saying x ~ y İfzy > 0. (a) Prove that is an equivalence relation (b) Determine all distinct equivalence classes of~ prove that your answer is correct.
Do both please will thumb up
Prove that n2 +1 > 2 for any positive integer n < 4. Use induction to prove: > 1.22 = (n-1)20+1 + 2,Vn e Z,n 1
1. Recall that x E R is positive (resp. negative) if x = (an) which is positively (resp textitnegatively) bounded away from 0. Prove the following LIM00 an for a Cauchy sequence n-oo (a) For any E R, exactly one of the following is true: x is positive, is negative, or x= 0 E R is positive if and only if -x is negative. (b) (c) If x, y E R are both positive, then x + y and xy...
Prove the following version of the division algorithm, which holds for both positive and negative divisors. Extended Division Algorithm: Let a and b be integers with b = 0. Then there exist unique integers q and r such that a = bq + r and 0 sr<|bl| [ Hint: Apply Theorem 1.1 when a divided by [b]. Then consider two cases (b >0 and b < 0) Explain the answer and visible for read
please post clear picture or solution.
Bonus question: 4 bonus marks] A positive integer r is called powerful if for all prime numbers P, p implies p | r. A positive integer z is called a perfect power if there exist a prime number p and a natural number n such that p". An Achilles number is one that is powerful but is not a perfect power. For example, 72 is an Achilles number. Prove that if a and b...
PLEASE DO BOTH
(5) (5 pts) Prove the identity (1) () = (x) (---), whenever n, r, and k are non-negative integers with rn andk Sr. (6) (5 pts) How many ways are there to select five unordered elements from a set with three elements when repetition is allowed?
7. Prove that for any positive real number r, if r is not an integer, then [x]+-1= 1
answer all parts please! :)
1. Recall that (an) which is positively (resp textitnegatively) bounded away from 0. Prove the following: eR is positive resp. negative) if x = LIMn0 an for a Cauchy sequence R, exactly one of the following is true (a) For any x is positive, r is negative, or (b) xRis positive if and only if -x is negative also positive (c) If ar, y E R are both positive, then r + y and ry...
k is a constant, it can be
positive or negative
3. Prove that the quantity s--k Σ P, In(R) r=1 is a maximum when P, = 1/n for all r. Do this by showing that S-kinn-0. Hint: The inequality In [1/(nP )] 1/(nP,) 1 will be helpful. TO arrive at an expression in which you can use this inequality, you will have to insert Ση_B-1 in an appropriate place. This is a problem which is easier not using calculus because...
Please do both questions. wrong answers will be given thumbs
down.
Question 7. Prove using the Division Lemma that Yn E Z, n3 n is divisible by 3 (any proof not using the Division Lemma will receive no credit). Question 8. Define a relation ~ on R \ {0} by saying x ~ y İfzy > 0. (a) Prove that is an equivalence relation (b) Determine all distinct equivalence classes of~ prove that your answer is correct.
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