Do all infinite sequences have the same number of terms?

1. When you can say two infinite sequences are the same size?

It depends what is meant by 'the size'. It could be that the size is the total length of the sequence or the distance between terms.

For the total length of the sequence, they're all the same size, as they all go on for infinity, but if size refers to the distance between the terms then if they start on the same number and each term is the same distance apart in both sequences, they will be the same size.

2. Can you give an example of two infinite sequences that are different sizes? If not, why not?

If the total length of the sequence is what 'size' mean then no because, as stated above, all sequences go on for ever. If, however, size means 'the distance between consecutive terms' then it is very easy. Of the sequences 2n and 3n, where n is every natural number, 2n produces more numbers before 100 (for example) than 3n because the terms aren't so far apart.

3. If an infinite sequence includes duplicates might it have only a finite number of different terms? If so give an example. If not explain why.

Alex's Sequence:
 
There are 2 robots. The first one, we shall call it A, takes a number, such as 4, multiplies it by 1, then passes it to another robot called B, who adds another number to it, like 16. Because A keeps churning out 4s and B keeps adding 16s to them, you always get 20.
 

Sarah's Sequence:
 
Get any number and add zero. That is term one. Then add zero to term one. That is term 2. Add zero to that and that is term 3 and so on.
 

Alex's Second Sequence:
 
Take a number then subtract the number from the number. You will always get zero.

4. Can you describe sequences that can't be programmed?

We cannot describe any but we are pretty sure that they do exist and may have something to do with irrational numbers.

5. Are there infinite sequences that have so many terms they can't be counted?

We have argued about what a sequence actually is and have come to the conclusion that it depends on what you mean by sequence. A sequence could be a list of number starting in a certain place, changing according to a certain rule each time. Or it could just be the rule, e.g. n+1. 

Yes, you can't count to the end, we agreed that there is no end, we are still not sure if there is a beginning.