## 45 Reputation

11 years, 285 days

## recursive sequence conjecture and proof...

Maple
• 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 ?

## sequence and generations...

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 ?

Maple

## abstract algebra...

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.