This sequence is called “Look and say sequence”. The sequence starts with 1 and for the next term, you need to ‘say what you see’ in the current term. i.e. how many of each numeral you see from left to right.
The fourth term is 1211. It reads as ‘one1, one 2. Two 1’.
So the fifth term is 111221.
This reads as ‘Three 1, two 2, one 1’.
So the sixth term is 312211.

~MetaHub Panel for Teachers’ Forum

1. The answer you have provided to your own question #1 is not the answer to the question you have asked. You have explained a procedural step to find the composite function f(g(x)) (when it exists) when f(x) and g(x) are given. But your question (which is not clearly stated), we think, is about the conditions under which a composite function exists. Assuming it is “What are the conditions required for f(g(x)) to exist?”, the answer is as follows:

          Conditions for f(g(x)) to exist are,

          * the range of g(x) must be a subset of the domain of f(x)

                Or

           *f(g(x)) must be defined with the domain of g(x) restricted so that its range is a subset of the domain of f(x).

     2. This has been already answerd in the response to #1.

     3. This question makes no sense. (what’s k?)

~MetaHub Panel for the Teachers’ Forum

What you should be teaching your students is not that “the sum of rational numbers is rational”, but, “the sum of two rational numbers is rational” which you can extend, using the principle of mathematical induction, to “the sum of a *finite* number of rational numbers is rational”.

It should be clear that this claim cannot be extended to *infinite* sums blindly. In fact, by definition, an infinite sum is actually the limit of a finite sum (partial sums).

“e” is a counterexample for this. The infinite sum of rational numbers need not necessarily be a rational number. 

~MetaHub Panel for Teachers’ Forum