if {5+2(6)^1/2}^n = I + f ; n,I are natural numbers; and f is greater than equal to 0,smaller than 1,(contd)?

Question

then I equals to what??
ANSWER: 1/ (1-f) - f
how do i get this answer?
pls help me!!! if the calculation is too diffucult to type, plz indicate the steps leading to the answer in words.
any help will be appreciated
thanks a lot.

then I is equal to....??
^ means 'raised to the power' example- if i say- (2^2) , it means-( 2 raised to the power of 2) = 4.
the answer precisely is-

(1-f+f^2)/(1-f)
if u think the answer is incorrect, then plz tell me the correct answer.

Answer 1

This problem is an exercise in conjugates.

5 + 2√6 = 9.8989794855...
Therefore, we can tell that (5 + 2√6)^n can always
be represented as I + f (an integer + a fraction).

5 - 2√6 = 0.1010205144...
Here, we can tell that (5 - 2√6)^n will always
be represented as a fraction less than 1.
Let's call this fraction, g.

We have, so far :
(5 + 2√6)^n = I + f
(5 - 2√6)^n = g

Now multiply them together to get :
(I + f)g = (5 + 2√6)^n * (5 - 2√6)^n
= [(5 + 2√6)(5 - 2√6)]^n
= (25 - 24)^n
= 1^n
= 1

So we have: (I + f)g = 1
Rearranging to find I gives: I = 1/g - f (EQUATION 1)

Now for the difficult part, "difficult" because intuitively, I feel
there must be an easier way. I've been working on this for
quite a while, but I just can't see the easy way out.

Let's look at (5 + 2√6)^n + (5 - 2√6)^n for various n.

1) (5 + 2√6)^1 + (5 - 2√6)^1 = 10
2) (5 + 2√6)^2 + (5 - 2√6)^2 = 98
3) (5 + 2√6)^3 + (5 - 2√6)^3 = 970
4) (5 + 2√6)^4 + (5 - 2√6)^4 = 9602

You'll notice they are all integers. This fact can be
ascertained by expanding the binomials and then
finding that all ODD values of the powers of 2√6 cancel,
thus leaving only integers to sum.

So we have established that:
(5 + 2√6)^n + (5 - 2√6)^n = J (where J is an integer)

But, (5 + 2√6)^n + (5 - 2√6)^n = I + f + g (from above formulae)

Therefore, I + f + g = J, or, f + g = J - I.

So we have fractions on the LHS and integers on the RHS.
The only way these fractions can be added to
equal an integer, is if f + g = 1. Thus, g = 1 - f.

Plugging this value in EQUATION 1 gives:
I = 1/(1 - f) - f, which is the answer you gave.

Edit: Sorry for mistake, but I've now fixed it - see above
where it now says "ODD values of the powers of 2√6".

Answer 2

The answer provided is incorrect. Or the statement of the problem is insufficient or incorrect. Say the answer provided is correct, then l + f = (1/(1 - f) - f) + f = 1/(1 - f). Then
(5 + 2sqrt(6))^n = 1/(1 - f) for any natural number n, which
is impossible. Please state the problem precisely.

Answer 3

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Answer 4

Wow. I don't see the reasoning here. I'd like to know as well.

Answer 5

i'm sorry, i just turned 13 i only know up to 11th grade math from my tutoring and i dont know what ^ means.