a little math question...MI helppppp

[rachmaninovian]

prove by MI (mathematical induction)

(3 + sqrt 5)^n - (3 - sqrt 5)^2 is a multiple of 2^n

first question, is there a problem with this question? should the expression be over sqrt 5?

2nd question, if so, how do I exactly go about doing it? I got stuck >.<


EDIT OOPPS SOZ LOL I TYPED WRONGLY -.-

 

[JBCM627]

prove by MI (mathematical induction)

(3 + sqrt 5)^n - (3 + sqrt 5)^2 is a multiple of 2^n

Good luck proving this...

When n=3, that expression = 26sqrt(5) + 58. Not a multiple of 2^n.

Edit:
Same problem with:
(3 + sqrt 5)^n - (3 - sqrt 5)^2 is a multiple of 2^n
When n=1, the expression is 7root(5) - 11

 

[brunson]

First off, if it works for any integer, n, then it must work fo 1. Plug in 1 for n and if it doesn't form an equality, you're done.

 

[jason9000]

I suspect that the problem statement is probably:
Prove, for all n, that 2^n divides (3+sqrt5)^n+(3-sqrt5)^n

 

[rachmaninovian]

thats why I thought that the question was wrong, and perhaps the whole expression should be over root 5. which makes the equation equal to 2, for n=1, thus fulfilling the first step of MI...

I was just stuck at the step after assuming it is true for n=k....this is the biggest problem =P