Question

Show that \[\frac{y^2+2y+2}{y^2-2y+2}\] is a natural number if y is a natural number.

Answer 1

I was following this wonderful closed thread: http://openstudy.com/users/mukushla#/updates/5044a5d6e4b0a71fb32b1265

\[\frac{y^2+2y+2}{y^2-2y+2} = k \] is factored into \[y^2(1-k) + 2y(1+k) + 2(1-k) = 0\]

and from there, it was said that since y is natural, so is the product of its solution also natural.

so, product of roots = 2. Hence the roots are y = 1 or 2...

but, although I follow the factorization, I do not understand how that helps show the roots are either 1 or 2... and also, how do we know that the fraction divides evenly into a natural number?

Answer 2

it doesn't work if \(n=3\) for example

Answer 3

from product of roots=2 and y is natural number
which 2 natural number gives product as 2?

Answer 4

\[\frac{3^2+2\times 3+2}{3^2-2\times 3+2}=\frac{17}{5}\]

Answer 5

I'm sorry, I don't understand which variable n is...

Answer 6

ok

Answer 7

ok \(y\)

Answer 8

it is not true, so i wouldn't waste time proving it was true

Answer 9

hmm, okay, I misunderstood the logic from the other thread... I see now that it isn't true. Thank you.