Today, J and K used this spreadsheet alongside these tasks to explore recurrence relations. The first two tasks were relatively straightforward, designed to help them gain familiarity with recurrence relations for a variety of linear and non-linear sequences. The third task was an exploration of recurrence relations of the form:
Below is their record of the first few values of u(1), k and d that they tried:
The example in the red box was the first sequence that tended to a limit. The examples in the green box are special cases. The breakthrough came with the first example in the blue box. Here is a screenshot of the spreadsheet for this example, to give you an idea of what they saw):
It was clear to them that the sequence was tending to some limit. At this point, K said: "k makes the limit." (see writing in blue box). They tested this by changing k to 9, as can be seen, presumably to test some conjecture not yet made explicit, before changing it back to 0.3 and then changing d instead (see also blue box). This confirmed to them that the limit was a function of k and d, but that it was the value of k itself that determined whether there was a limit or not.
K then made the conjecture that k must be <1, and J mentioned that it must also be greater than -1. This can be seen in the red box in the next sheet they worked on:
The numbers in the red box (-100, -2, -1.3) are some of the values of k they used to test the -1 conjecture. K was excited about the boundary example in the green box, which alternates between 0 and 1.
Then they only altered k for values between 0 and 1 (keeping the first term and d constant), and looked at the changing values of the limit L. They could see there was some connection between the value of k and L, but could not derive it, at least in the time available. I think it would be very difficult and perhaps not that productive to discover how to work out L, I described an algebraic approach for finding L, using some of their examples to illustrate the method.
It felt as though this approach, of using a spreadsheet along some explorative tasks, was a nice way of introducing recurrence relations and the idea that some of them tend to a limit. We will look at graphical representations tomorrow.