if x and n are positive integers....

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November 6th 2009, 08:24 PM
everydayangel

if x and n are positive integers....

I'm stumped on this.. I don't remember this stuff and I need help. Not even sure it is in the right forum, but my Trig teacher gave us this on a worksheet so I tried my luck..
(Moderator - looked for a mod or admin online but none on to ask where to post this - please move to correct forum for me, if not in correct one.. thanks) (Hi)

If x and n are positive integers with x > n and Xⁿ - Xⁿ⁻¹ - Xⁿ⁻² = 2009, find x + n.

Choices:
A - 10
B - 11
C - 12
D - 13
E - 14

If anyone is able to answer this, can you explain how you got your answer so I can see what I should be doing here? I really appreciate it. Thanks..

Ang
(Angel)
November 6th 2009, 09:27 PM
Gusbob

Interesting Question. I have a solution that doesn't involve trig. There may be a way doing this using trig, but I'm not sure how.

First off we have $x^n - x^{n-1} - x^{n-2} = 2009$

Factorising $x^{n-2}$ gives us

$x^{n-2} (x^2 - x - 1) = 2009$

Since x is a positive integer, $x^2 - x - 1$ is also an integer. This means $x^{n-2}$ must also be an integer (but only if n > 1)

Now we note that the prime factorisation of $2009 = 7^2 \cdot 41$ .

Because $x^{n-2}$ is an integer only if n > 1, we can rule out the prime factor 41 as a possible value for x (it doesn't have a power > 1). This means x = 7. Logically, $x^2 - x - 1$ is 41 (substitute 7 into this to prove it to yourself)

Now divide both sides by 41 and substitute x = 7 to get $7^{n-2} = 49$

Clearly n = 4.

So we now we have x = 7 and n = 4. The answer to the question is B - 11.
November 6th 2009, 11:07 PM
CaptainBlack

Quote:

Originally Posted by everydayangel

I'm stumped on this.. I don't remember this stuff and I need help. Not even sure it is in the right forum, but my Trig teacher gave us this on a worksheet so I tried my luck..
(Moderator - looked for a mod or admin online but none on to ask where to post this - please move to correct forum for me, if not in correct one.. thanks) (Hi)

If x and n are positive integers with x > n and Xⁿ - Xⁿ⁻¹ - Xⁿ⁻² = 2009, find x + n.

Choices:
A - 10
B - 11
C - 12
D - 13
E - 14

If anyone is able to answer this, can you explain how you got your answer so I can see what I should be doing here? I really appreciate it. Thanks..

Ang
(Angel)

Try exhaustive search, start with x=14 ... 2, n=x-1 ... 1

CB