On first viewing, the prompt seems closed. Two expressions give linear sequences from which the common terms are combined to make a third sequence. The expression for the nth term of the new sequence is then deduced. However, this inquiry has generated a variety of questions about sequences. It has also led to highly focussed classes who become engrossed in trying to produce a general rule that connects the first two expressions to the third one.
The prompt teaches students an important lesson - there is not always a neat solution, or at least one that can be arrived at through spotting patterns. Indeed, at times, each new case seems to contradict a generalisation from the previous ones. Students can begin to appreciate that explaining the inner workings of one case is often better than producing more examples.
Classes have taken this prompt in different directions. At the start, students often aim to reproduce the procedure in the prompt with their own examples, developing fluency in finding an expression for the nth term as they do so. After that, different pathways have developed, including a group of year 8 students who looked at intersecting quadratic sequences using a spreadsheet (below). Another pathway involves inquiring into combinations of even and odd numbers in the first two expressions.
Resource sheets On sheet 1, the first and fourth pairs give the same expression for the nth term of the new sequence. This has led to further inquiry into whether there are more pairs that lead to 15n - 2. Odds and evens sheets Position-to-term rules and term-to-term rules. Initially, students have distinguished between the pairs that can and cannot produce an intersecting sequence. The Inquiry Maths session at the 2015 conference of the Association of Teachers of Mathematics was based on the intersecting sequences prompt. You can read a report of the session here.