Given a real number x and a positive integer k, determine the number of multiplications used to find x^(2^k) starting with x and successively squaring (to find x^2, x^4, and so on). Is this a more efficient way to find x^(2^k) than by multiplying x by itself the appropriate number of times?

Consider the polynomial p(x)=x^4−4x^3+6x^2−4x+1. Let d be the value of p(1.00055) evaluated using Horner's method in MATLAB. Give the value of d×10^13 rounded to 4 decimal places.

7. For n 2 1, in how many out of the n! permutations T = (T(1), 7(2),..., 7 (n)) of the numbers {1, 2, ..., n} the value of 7(i) is either i – 1, or i, or i +1 for all 1 < i < n? Example: The permutation (21354) follows the rules while the permutation (21534) does not because 7(3) = 5. Hint: Find the answer for small n by checking all the permutations and then find the recursive formula depending on the possible values for 1(n).

We have learned the algorithm that solves the Closest pair problem in 2D in Θ(n log n) time. (Closest pair problem in 2D: Given n points in the 2D plane, find a pair with smallest Euclidean distance between them.) Give an algorithm that solves the Closest pair problem in 3D in Θ(n log n) time. (Closest pair problem in 3D: Given n points in the 3D space, find a pair with smallest Euclidean distance between them.)

2. Prove that 8"-3" is evenly divisible by 5 for all natural numbers n.

Prove that the sum of the first n odd positive integers is n2. In other words, show that 1 + 3 + 5 + .... + (2n + 1) = (n + 1)2 for all n ∈ N.

suppose a,b, and c are odd integers. Prove that a+b+c is odd

Let n be an integer, use the definition of even to prove that 18n + 2 is even. 18n + 2 = Since X is an integer, is also an integer. Since 18n +2 equals 2 times an integer, 18n + 2 is even.