Graph of a sequence, monotonic and bounded
Draw a sequence as points (n, a_n): an arithmetic one lies on a line, a geometric one on a curve. Then decide whether a sequence always grows and whether it stays below a limit.
- France: grade 11
- Syria: grade 11
- Ukraine: grade 9
- Russia: grade 9
Guide
A sequence is a function of n = 1, 2, 3, ..., so its graph is a set of separate points (n, a_n). Do not join them with a line.
The points of an arithmetic sequence lie on a straight line; the points of a geometric sequence with ratio q > 1 lie on a curve that climbs ever more steeply.
A sequence is increasing if each term is bigger than the one before, and decreasing if smaller. It is bounded above if all terms stay below some number, and bounded below if all stay above some number. This is advanced-level material.
Common slips: reading n instead of the term on the vertical axis, adding the same amount to a geometric sequence, and calling every increasing sequence unbounded.