 Hong Kong
Stage 4 - Stage 5

# Graphs and Tables - Recurrence Relations

Lesson

The terms of any sequence that are generated by Recurrence formulae can be easily tabulated and graphed.

The graph of a recurrence relation can provide a visual understanding of the way the sequence behaves. For instance we may be able to see that that terms are always increasing or always decreasing in size. We might see that the terms appear to be approaching a limiting size. We might see the that the sequence starts to become erratic an unpredictable. We might also see that the terms change linearly, or as a quadratic curve, or as some other known form, and this might provide a hint as to a possible explicit form of the sequence.

The graph could take a number of forms. For example you might use a series of discrete points, or perhaps a histogram, or some other type of graph. The important point to be made is that the graph should highlight  the trend exhibited by the sequence, and so the vertical axis scale may need to be adjusted to see this, particularly if part of the graph climbs or falls rapidly.

Our first example concerns a stockbroker's claim that  $\$200$$200 can be turned into a \1000$$1000 in $10$10 months. He claims that he can make $30%$30% a month on the investment, for the cost of $\$35$$35 a month. Such a claim seems incredulous, but given the percentage earnings and the monthly fee stated, we can form a recurrence relation as: U_{n+1}=1.3U_n-35,U_1=200Un+1=1.3Un35,U1=200 Note that the coefficient 1.31.3 is equivalent to 130%130%, which is a 30%30% increase on the previous month's amount as shown by the formula. The subtraction of \35$$35 represents the monthly fee.

We can tabulate the size of the investment at the beginning of the month for the first 10 months as follows, using month numbers and rounded whole dollars:

M 1 2 3 4 5 6 7 8 9 10