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Grandad · #5 July 5th, 2009, 03:07 AM

Hello matsci0000

Thanks for showing us your working. This part is pretty well OK, except for where I've commented.

Quote:

Originally Posted by matsci0000

I've solved the first part of the question in the following way(with the help of second hypothesis):

Given :x^2-(p+1)x+1=0 and alpha and beta are the roots .
so, alpha+beta = p+1 ..............(1)

Correct. But you don't need this bit:

Quote:

and alpha+beta is greater than or equal to 4.............(2){since it is given that p>=3}

Quote:

alpha x beta =1 ................(3)

Correct. Notice that this now shows that $\alpha+\beta$ and $\alpha\beta$ are integers.

Quote:

STEP1 -- To prove that P(1) is an integer.
alpha^1+beta^1 =alpha+beta -------> is an integer. [from (1)]

To prove that P(2) is an integer.
P(2)=alpha^2+beta^2=(alpha+beta)^2- 2alpha .beta

Correct, and this is all you need, in order to show that $\alpha^2+\beta^2$ is an integer. So you don't need this bit:

Quote:

{(alpha+beta)^2 is >=16 [from 2] , 2alpha .beta =2 [from 3]
so, the value of alpha^2+beta^2=(alpha+beta)^2- 2alpha .beta becomes >=14 and hence it is an integer.}

Now for the next part:

Quote:

STEP 2 INDUCTION ASSUMPTION
--------------------------------
Let P(k) be an integer.
so, alpha^k +beta^k be an integer.
Let P(k-1) be an integer.
so,alpha^(k-1) +beta^(k-1) be an integer.

Step3 -- To prove that P(k+1) is an integer.
P(k)=alpha^(k+1) +beta^(k+1)

You mean $P(k+1)$ here, of course.

Quote:

=(alpha^k +beta^k )(alpha+beta) - alpha^k.beta-beta^k.alpha
=(alpha^k +beta^k )(alpha+beta) - alpha .beta{alpha^(k-1)+beta^(k-1)}

This is fine. In other words

$P(k+1) = P(k)(\alpha+\beta)-\alpha\beta P(k-1)$

So this, together with your assumptions that $P(k)$ and $P(k-1)$ are integers and the fact that $\alpha + \beta$ and $\alpha\beta$ are both integers is all you need to show that $P(k+1)$ is an integer.

Since you've already established that $P(1)$ and $P(2)$ are integers, this completes the proof.

So you don't need this bit:

Quote:

USING-
alpha+beta is greater than or equal to 4.............(2)
alpha x beta =1 ................(3)
(alpha^k +beta^k )(alpha+beta) >=4 [from above 2 relations]
and alpha^(k-1)+beta^(k-1)

I haven't time now to look at your proof of part 2. If no-one else has commented on it, I'll do so later.

Grandad