• here is an exercise I got from a text book                                                                                                              calculate the first 10 terms of the following sequence :                                                                                              

                                                                                                                                                             u[n+1]=1/2(u[n]+2/u[n]) n>=0                                                                                                                          

• estimate the differences u[3]-sqrt(2) , u[4]-sqrt(2), u[5]-sqrt(2), and u[6]-sqrt(2) with a precision of 50 numbers                                    
• what can we conjecture about the sequence ?
• how to prove that conjecture with MAPLE ?

Egor has two parents, four grandparents, and so on.
Write an explicit formula and a recursive formula for the number of ancestors Egor has if we go back n generations.                

what would be the figure back to 25 generations ?

Let (G, ·) be a group and X any set. Let F be the set of functions with domain

X and range G. Define a binary operation ∗ on F by (f ∗ g)(x) := f(x) · g(x). Is

prove that this is so.

Yes, (F, ∗) is a group.

prove it.

Exercise Prove that (-1)u = - u in any vector space. Note that (-1)u means the number -1 is multiplied to the vector u, and - u means the negative vector in the fourth property of the definition of vector spaces.

Answer

Exercise Prove that (a₁u₁ + a₂u₂) + (b₁u₁ + b₂u₂) = (a₁ + b₁)u₁ + (a₂ + b₂)u₂ in any vector space.

Answer

Exercise Give a detailed reason why, in any vector space,

• u + v = 0 ⇒ u = - v.

• 3u + 2v - 4w = 0 ⇒ v = - 3/2 u + 2w.