Consider the sequence: x₀=1/6 and x_n+1 = 2x_n- 3x_n² | for all natural numbers n.

Show:

a) x_n< 1/3 for all n.

b) x_n>0 for all n.

Hint. Use induction.

c) show x_n isincreasing.

d) calculate the limit.

(a) Let a > 1 be an integer. Prove that any composite divisor of a − 1 is a pseudoprime of base a.

(b) Suppose, for some m, than n divides a^(m − 1) and n ≡ 1 (mod m). Prove that if n is composite, then n is a pseudoprime of base a.

(c) Use (b) to give two examples pseudoprimes of base a with a = 2 and a = 3 (hint: take m = 2k to be an even number).

Explain how technology can be used to address one specific misconception in secondary-level geometry. (When thinking of a misconception, you want to mention something specific within a geometry course where a student may make an error).

Show that, in n-dimensional space, any n + 1 vectors are linearly dependent.

HINT: Given n+1 vectors, where each vector has n components, write out the equations that determine whether these vectors are linearly dependent or not. Show that these equations constitute a system of n linear homogeneous equations with n + 1 unknowns. What do you know about the possible solutions to such a system of equations?

Use Classic Runge-Kutta method with h = 1 to solve the system y” - y’ - 6y = 0, y(0) = 2, y’(0) = 3 on [0,1]

Show that the eigenfunctions un(x) are orthogonal .

In Hilbert's Axioms, all of the axioms of connection are independent of each other

5) Let the function f : ℝ³ → ℝ³ be given by f(x, y, z) = (2x + 2y, 2y + 2z, z + x).

a) Prove that f is one to one and onto
b) Find the inverse of f, i.e., f⁻¹.

Consider a square matrix A such that Ker(A2 ) = Ker(A3 ). Is Ker(A3 ) = Ker(A4 ). Explain your reasoning.

use muller's method to find the roots of the equation f(x) = sin x - x/2 =0 near x=2

How do you recognize in which situations the idea of strong induction might be useful?