Question

Salut,
J'ai un exercice:
Montrer par récurrence que x^n+1/x^n=2cos(nx) n appartient N/{0}. Sachant que x+1/x=2cosx

Answer 1

hello :)

Answer 2

do you mean :

show that\[x^n+\frac{1}{x^n}=2 \cos (n \theta)\]if\[x+\frac{1}{x}=2 \cos \theta\]?

Answer 3

oui

Answer 4

yes !

Answer 5

Who can help me ?

Answer 6

is n a positive integer? and will proof by mathematical induction do?

Answer 7

yes

Answer 8

let suppose statement holds for \(n\) \[x^n+\frac{1}{x^n}=2 \cos (n \theta)\]so using mathematical induction we must prove it holds also for \(n+1\) or in mathematical words\[x^{n+1}+\frac{1}{x^{n+1}}=2 \cos ((n+1) \theta)\]

Answer 9

yes

Answer 10

very well :)

Answer 11

now i give u a hint

Answer 12

are u supposed to use mathematical induction for sure?

Answer 13

yes

Answer 14

can you continue the calculation

Answer 15

well yes

Answer 16

using 2 statement that we know they are right\[\left(x+\frac{1}{x} \right) \left(x^n+\frac{1}{x^n} \right)=4 \cos \theta \cos (n\theta)\]am i right?

Answer 17

yes

Answer 18

continue pleaze

Answer 19

ok distribute the LHS \[x^{n+1}+\frac{1}{x^{n+1}}+x^{n-1}+\frac{1}{x^{n-1}}=4 \cos \theta \cos (n\theta)\]note that we use strong induction so\[x^{n-1}+\frac{1}{x^{n-1}}=2 \cos((n-1)\theta)\]put this in the latest equation u will get\[x^{n+1}+\frac{1}{x^{n+1}}+2 \cos((n-1)\theta)=4 \cos \theta \cos (n\theta)\]\[x^{n+1}+\frac{1}{x^{n+1}}=4 \cos \theta \cos (n\theta)-2 \cos((n-1)\theta)\]now you should prove RHS is equal to \(2 \cos((n+1)\theta)\) for completing the proof

Answer 20

makes sense?

Answer 21

can you continue pleaze ?

Answer 22

using formula \[\cos A \cos B=\frac{1}{2}(\cos(A+B)+\cos(A-B))\]we will have\[4 \cos \theta \cos (n\theta)-2 \cos((n-1)\theta)=2(\cos((n+1)\theta)+\cos((n-1)\theta)-2 \cos((n-1)\theta)\]\[4 \cos \theta \cos (n\theta)-2 \cos((n-1)\theta)=2 \cos((n+1)\theta)\]

Answer 23

thanks !!

Answer 24

no problem

Answer 25

actually mathematical induction is not what i interested in for this problem

Answer 26

i give you a solution for fun :)

Answer 27

do you know complex numbers?

Answer 28

discuss the values ​​of these two equations:
1) x ²-2xsina-cos ² a = 0
2) ² x ²-2x + cos a = 0

Answer 29

pleaze this exercice

Answer 30

post it in open questions, i gotta go :)