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:

Here are the questions that I invited them to explore:

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.