Possible groupings

Sort these infinite sequences into groups.

  • How will you choose to define the groups?
  • Are there some sequences which belong to more than one group?

Here is one possible grouping. How do you think this grouping was constructed?

venn diagram of grouping with APs, GPs, periodic, quadratic and other

Some didn’t seem to fit in any of these groups.

Why can we think of the sequence \(1,1,1,1,...\) as both an arithmetic sequence and a geometric sequence? Are there any other sequences that are both arithmetic and geometric?

Can we find a sequence that fits in the empty parts of this grouping diagram?


Here is another possible grouping. How do you think this one was constructed?

venn diagram of grouping with periodic, convergent, divergent and other seq

If we plotted the graphs of the convergent sequences, what would they have in common?

If we plotted the graphs of the divergent sequences, what would they have in common?

Can we find a sequence that fits in the empty parts of this grouping diagram?


And another… how do you think this one was constructed?

venn diagram of grouping with increasing, decreasing, periodic and other seq

One didn’t seem to fit in any of these groups.

Why do you think our groupings are labelled strictly increasing and strictly decreasing sequences? (It might help to think about increasing and decreasing functions.)

How would the grouping diagram have changed if we had not used the word strictly in our labelling of our groupings?