Icm(a, b) = ab if and only if a and b are relatively prime.

Q: 2) Prove by induction on n that, for all positive integers n: Σ 6(5n6" – 6" + 1) i6 25

A:

Q: gcd(a, b) = 1 → a and b are relatively prime. %3D

A: Relatively prime numbers: Any two integers a and b are said to be relatively prime if and only if…

Q: n2 +2 2 3n for n = 1,2,3,.

A:

Q: 2. If b and c are relatively prime and a / bc then a / b or a /c.

A: First Method: Since b and c are prime: gcd (b,c) =1 Then: gcd (ab, ac) is equal to "a" Then: Let for…

Q: 5.2.8 suppose that a store offers gift certificates and denominations of $25 and $40. Determine the…

A: The given denominations are $25 and $40.Here, the given denominations are multiples of 5.The amounts…

Q: Find the closed form solution of recurrence relation defined by T(1) = 2,T(n) = 2n · T(n – 1), n > 1…

A:

Q: ) Find the formula for bn given by the recurrence bn+1 = 3bn + 2n, bo = 1

A: As per the guidelines I am answering only first question. i.e. (d) bn+1=3bn+2n, b0=1

Q: Prove that 2n > n2 for n ≥ 5. Show that the formula is true for n = 5 and then use step 2 of…

A: Given n≥5

Q: 20. LCI T1 UC u 21. Give a proof using the Pigeonhole Principle: (a) The Italian village of Solomeo…

A:

Q: Icm(a,b) = ab if and only if a and b are relatively prime.

A: We know the relation between product of the two numbers their L.C.M and G.C.D Let a and b two…

Q: By Principle of Mathematical Induction prove 6n 6

A:

Q: With the help of principle of mathematical induction, prove that 1 * 2 + 3 * 4 + 5 * 6 +…….+…

A:

Q: For neN, is f (n) = 3.2" - 3 a solution to the recurrence relation An = 3an-1-2an-2?

A: For the recurrence relation an = 3an - 1 - 2an - 2  we have f(n) = 3•2n - 3 for all n in N as a…

Q: Find the formula for bn given by the recurrence bn+1 = 3bn + 2n, bo = 1

A:

Q: Solve the following DE Using Method of Undetermined coefficients; applying both (a) Superposition…

A:

Q: Find the solution to the following Ihcc recurrence: An 2an-1 for n > 2 with the initial condition a1…

A:

Q: Use mathematical induction to show that Σ (4k – 1) = n(2n + 1) %3D k=1

A: To prove the given statement by using mathematical induction.

Q: Show that 1+3/ -5 is irreducible which is not prime in ZlV-5]

A: We know that ,  ℤ-5 = a + b-5 / a , b ∈ ℤ We need to show that , 1 + 3-5  is irreducible but not…

Q: A shop gives gift cards in nominal 25 and 40k. determine the possible total amount that can be…

A: Proof at first, the amounts seem semi-random:

Q: Show that 5n + 3 and 7n + 4 are relatively prime for all n.

A:

Q: prove using induction: II ( 1 1 1 П 2k +1 2k+2, for n 20 (2n + 2)! k-0

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Q: 1 (b) Prove by induction that (3i – 2)² = ;n(6n² – 3n – 1) for n 2 1

A: We have to prove ∑i=1n3i-22=12n6n2-3n-1 for n≥1 by using mathematical induction.

Q: Use any method to deter (a) 13 The series 8.

A:

Q: If 2n + 1 is prime for some n ≥ 1, prove that n is a power of 2.

A: We have given 2n +1 is prime for some n ≥ 1 We have to prove that n is a power of 2.

Q: Use mathematical induction to show that, Vn > 3, 2n² + 1 > 5n.

A:

Q: If gcd(m,p) = 1 and p is prime, show that gcd(m,pk ) = 1 for all k≥ 1.

A: To show, If gcd(m, p)=1 and p is prime, show that gcd(m, pk) = 1 for all k≥1.

Q: If possible, let berational. v2 a Then, there exist positive co -primes a andb such that - -

A:

Q: The global error in a Runge-Kutta 4th order method is "big O" of 04 O 4h 4 Oh4

A:

Q: Theorem 11.3. If p and q = 2p+1: primes, then either q | M, or q| M,+2, but are not both.

A: As per Bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: Σ- n+ 1) (2n + 1), n2 1 i=1

A: Induction method says that if n=1is true then assume that n=k then prove that the statement is true…

Q: 2. Prove that Qn [2, 3] ~ N.

A: Two sets A and B have the same cardinality if there exists a bijection (onto , one-to-one…

Q: If gcd(a, b) = d, then a and- are not relatively prime. d %3D

A:

Q: If n 2 1, then the Fermat number Fn 22" +1 is prime if and only if %3D 3(F.-1)/2 = -1 mod (Fn).

A:

Q: Show that 6n – 1 and 12n+1 are relatively prime for any n e Z.

A:

Q: Use mathematical induction to prove that 1+ 2" 1. (Upload your solution.)

A:

Q: 6. Use mathematical induction to show that, Vn > 7, n! > 3".

A:

Q: n>1 Problem 4. Prove by induction that, for every geq2 1 n +1 - - - 32 n2 2n

A:

Q: 4) Prove by induction on n that, for all positive integers n: 6(5n6" – 6" + 1 i6' = %3D 25 i=

A: # Prove  ∑1=1n i 6 i=65n.6n-6n+125      Let Pn be the proposition given above Base case:-…

Q: 10. If you spin the spinner 70 times, how many times would you expect the spinner to land on a…

A: Here total section of spines are 5 in which we want to find spinner to which land on section with 2.

Q: Prove by the Principle of Mathematical Induction that 1(1!) + 2(2!) + 3(3!) + .. +n(n!) = (n+1)!- 1…

A:

Q: Exercise 9.3.2. Show that if p and q are primes, then the number of positive integers less than pq…

A:

Q: if gt2) Use induction to prove: 1 ΣΤΗ Σ 2 2n n > 0. hint: 1 2n 2 2n+1

A:

Q: 1) Guess a formula for the sum 1-3+5-7+..+ (-1)*-(2n – 1) =(-1)-(2i – 1) i=1 and prove your fornmula…

A: The given sum is  1-3+5-7+...+-1n-12n-1=∑i=1n-1i-12i-1

Q: Show that if a and b are relatively prime then (a+2b,2a+b)=1 or 3.

A:

Q: (Zp-f0?),where P Prime number is an Abellan gooup.

A: We have to verify whether the given statement is true or not .

Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.