Which of the following is the algorithm of Newton's Method for finding x3 = N when N is a real number?

Q: The sum of any two integers of the from 4k + 1is always: Od Zero Even prime

A: Sum of two integers of the form 4k+1

Q: Prove that n2 > 2n + 1 for n ≥ 3. Show that the formula is true for n = 3 and then use step 2 of…

A: To show that n2 > 2n + 1 for n ≥ 3 using mathematical induction.

Q: Show that (2i – 1) (2n)! for every positive integer n!2" n. i=l

A: We have to solve given problem:

Q: (b) Prove by induction that (3i – 2)² = n(6n² – 3n – 1) for n 21 %3D i=1

A: Prove the equation ∑i=1n3i-22=12n6n2-3n-1 for all n≥1. Prove the equation for n=1. Substitute n=1…

Q: (5n)! 40"-n! is natural number for every natural number n. Prove that

A:

Q: Prove by induction that for all n > 2, Ln = Fn-1 + Fn+1 Where La denotes the nth Lucas number and Fn…

A:

Q: 62. Prove part (b) of Theorem 4.4 in the case when n is odd: If n(n+1) n is a positive odd integer,…

A: ∑k=1nk=n(n+1)2 , n is odd integer

Q: Define a sequence of real numbers as follows: Let x1 = 1, and then for ever n ∈ ℕ, define xn+1 = (xn…

A: Given : a sequence of real numbers such that         x1=1 and then for every n∈ℕ, define xn+1=xn2+1…

Q: Q2:Write an Algorithm to evaluate the following equations: X4 + 2! x2 x6 X8 + 6! 1.Cos(X) = 1 – -- -…

A:

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: Use induction to prove (2k – 1) = n² – 4 k=3 for all positive integers n2 3.

A:

Q: Prove by mathematical induction that for n E N, 4r2 - 1 2n + 1 T=1

A:

Q: Show that n² + 4n + 14 is big-Oh (n²)

A: Take k=4 and C=6

Q: Find a number n e N such that 2" > 102000 and 2" has among its 2000 last digits at least 1000…

A: We will use logarithm to solve the problem

Q: 5. Prove that the equation ø(n) = ¢(n+ 2) is satisfied by n = 2(2p – 1) whenever p and 2p – 1 are…

A: Euler totient function formula: p is prime then ϕ(p)=p-1 ϕ(pk)=pk-pk-1     ∀k∈ℕ   If gcd(a,b)=1…

Q: Find k such that, 2 det 4 li0 -8 -5] -1 -7 = 510 k 0 .

A: Given det2-8-54-1-710k0 =510 we have to find k

Q: 15. In a computation with Euler's method, the following results are obtain- ed with various step…

A: Consider the given data, h y h0=2-2 2.44141 h1=2-3 2.56578 h2=2-4 2.63793 Now, calculate…

Q: Use Euclidean algorithm to obtain x and y satisfying the equation of (5 036, 2 601) = 5 036x + 2…

A:

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: numerically greatest

A:

Q: 15. Find algorithms for getting the third and fourth roots of N that are similar to that in Exercise…

A:

Q: Find ln(9.2), using quadratic interpolation, from ln(9.0) = 2.19722, ln(9.5) = 2.25129, ln(10) =…

A: Given ,  ln(9.0) = 2.19722, ln(9.5) = 2.25129, ln(10) = 2.30259, and ln(11) = 2.39790  We have to…

Q: Solve the given LP Min model using Simplex Algorithm. Min z = 4x1 – X2 s.t. 2х, + х, < 8 X2 55 X1 -…

A: Min Z = 4 x1 - x2 subject to 2x1 + x2 ≤ 8 x2 ≤ 5 x1 - x2 ≤ 4 and x1,x2≥0;After introducing slack…

Q: п 6. Prove that E(8i – 5) = 4n² -n for every positive integer n. п i=1

A: # Given: sum(8i-5) =4n^2-n  i=1,2,3,.....n  We have to prove the above equation??

Q: 4. The "divide and average" method, an old-time mthod for approximating the square root of any…

A:

Q: Solve for d, 19d ≡ 1 mod (37422000) using the "Extended Euclid's Algorithm"

A: The solution is given by using Extended Euclid's Algorithm as follows

Q: Use Euclidean Algorithm to obtain x and y satisfying the (25 000, 2 460) = 25 000x + 2 460y

A:

Q: 5. Find the generating function to determine the number of ways to pick k objects from 2n objects…

A: Given: When duplicates are allowed and the (2i – 1)th and (2i)th objects occur the same number of…

Q: Prove by induction  that the sum of the first n even integers starting at 2 is n(n+1).  (That is,…

A:

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: 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: п Find a formula for II (1- ) : and verify. k2 k=2

A: We will expand the terms and check which terms are getting cancel each other

Q: Find all pairs of consecutive odd natural numbers, both of which are larger than 10, such that their…

A:

Q: In this question you will use strong induction to prove that your new algorithm is very efficient.…

A: Inductive Step:           For the inductive step we can assume some arbitrary number…

Q: 10. Prove by mathematical induction that E k (k + 1) = n (n + 1)(n + 2)/3 1sksn 11. Prove by…

A: Given expression: (10)                                                   First , we check if it is…

Q: Verify by mathematical induction Ž (2i – 1)° = n°(2n² – 1) for alln2 1. - i=1

A:

Q: (b) Prove by Induction that (3i – 1)(3i +2) = 3n +6n2 +n for n21 i=1

A:

Q: part ii and iii, thank you

A: According to pascal law,

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

A:

Q: 5. In a computation with Euler's method, the following results are obtain- ed with various step…

A:

Q: Prove by induction:  gn=gn-1+gn-2+gn-4   for n greater than or equal to 4, with g0=1, g1=1, g2=2,…

A:

Q: (5n)! 40 n! is natural number for every natural number n. Prove that

A:

Q: (a) Find two consecutive primes that differ by at least 10. (b) Prove that there are arbitrarily…

A: According to the given information, for part (a) it is required to find the two find the two…

Q: What is the largest n for which one can solve within a day using an algorithm that requires f (n)…

A: Consider the provided question, Hello. Since your question has multiple sub-parts, we will solve…

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 the sum of this: pi/(3sqrt(3))=(∑∞ n=0 ((-1)n (1/2)3n+1))/(3n+1) + ∑∞ n=0  ((-1)n…

A:

Q: 5.6 Determine the positive real root of In (x4) = 0.7 (a) graphi- cally, (b) using three iterations…

A: Bisection method: Let f be continuous function in [a,b] In this method we take two initial guess…

Q: Make a table of values of the numbers 10n – 1 and 10n + 1 for 1 < n < 20. Circle any numbers that…

A: Table of values for 10n-1 and 10n+1 for n between 1 and 20. n 10n-1 10n+1 1 9 11…

Q: Solve this recurrence relation together with the initial conditions given an=2an-1-an-2 for n≥2…

A: We will first write the characteristic equation of the given homogenous recurrence relation and find…