Math Is Fun Forum

The only problem I can see is the cheif who can answer truthfully or falsely.  It's possible that he would ansewer "true" if you asked him if 1+1=2, but then he lies to your question, and would answer "no".

I don't understand your question.  Are you trying to find the base value of T, like T(1)?  Or are you trying to find a non-recursive definition of T(n)?

If it's the first, the problem is easy.  You know T(3) = 3 x T(2) x 2^2 = 3 x 4 x T(2) = 36, so T(2) = 3.  Using this same method you can find that T(1) = 1/2.

If you need a non-recursive definition of T(n), try looking backwards.  What I mean is to do something like this:

The pattern here is pretty easy to see.  We have 3 raised to the power of n-1 at the start.  We then have 2 raised to the power of n^2 - (sum from i = 1 to n of i).  This is then multiplied by our base T(1), which we know from above is 1/2.  This leaves us with

Let's check that this is right.  For T(1) we get 3^0 x 2^(-1) = 1/2.  For T(2) we get 3^1 x 2^0 = 3.  For T(3) we get 3^2 x 2^2 = 36.  And finally, T(4) = 3^3 x 2^5 = 864, all of which are what we expected.

x + y > 2, x - y < 2
x > 2 - y, x < 2 + y
->
2 - y < 2 + y
->
0 < 2y
->
y > 0

The 'solution' would be y > 0, 2 - y < x < 2 + y (alternatively, |x - 2| < y). I've never heard of such a thing mikau, would you mind posting your method?

It is John.  6(a+1)(a-1)(a-1) = 6(a-1)(a^2-1), which is what luca said.

Here's some information that may be helpful.  Check this Wikipedia link:

http://en.wikipedia.org/wiki/Infinity#Cardinality_of_the_continuum

This paragraph in particular is of interest:

Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space. These results are highly counterintuitive, because they imply that there exist proper subsets of an infinite set S that have the same size as S.

A proper subset is a subset that has fewer elements than it's parent.  For example, let A be a set containing {1, 2, 3, 4}, B containing {1, 2, 3}, and C containing {1, 2, 3, 4}.  B is a subset of A because all of it's elements are contained in A.  It is also a proper subset because it does not contain every element of A, namely 4.  C is a subset of A because all of its members are contained in A.  However, C is not a proper subset of A because there are no elements in A that are not in C.

Now, re-read the Wikipedia quote.  There exists proper subsets of S (S being any infinite set, countable or otherwise) that have the same size as S.  One such example would be A = {1, 2, 3, 4, 5, ...} and B = {2, 3, 4, 5, 6, ...}.  B is clearly a proper subset of A, but it can also be shown that they have the same size.  It's counterintuitive, but it's been rigorously proven.  This is another Wikipedia page that may be helpful to study:

http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel

Judging by the above responses you originally accidentally typed the denominator as 3x^2 - 2, when it should be 3x^3 - 2.  Assuming that you now have the correct expression, follow LuisDog's advice again, but this time with x^3:

The limits of each of the terms with an x in the denominator is 0 as x approaches infinity, so they can be safely removed from consideration.  We're left with

which has no x, so the limit is just the expression itself, -1/3.

Use the rule log(a) + log(b) = log(ab).  Applying this rule to your example you'd get log(x) + log(8) = log(8x) = 2.  It should be fairly straightforward to solve from there.