I love sequences. I like their sometimes predictable nature and I love their sometimes unpredictable nature. I thought I’d share some ideas within the topic of sequences to help students deepen their understanding and perhaps pique their interest. Students study sequences at different depths throughout secondary school and so the ideas below could work for different classes at different stages. Please let me know if you have any success with any of them!
Using the OEIS
The Online Encyclopaedia of Integer Sequences (OEIS) is a database of famous (and not so famous!) sequences, available at oeis.org. It contains over 320,000 sequences and it’s really easy to search for a sequence and find out more information about sequences. Recently, I set my Year 8 class a challenge to invent a new (as yet undiscovered!) sequence for the OEIS. The idea of being the first to come up with a sequence and to have it entered in the OEIS under their name was very appealing to the students. I had some really interesting (and bizarre!) sequences and it certainly beat marking a ‘normal’ piece of homework! Students came up with sequences that led to discussions about triangular numbers, factorials and more. A couple of examples:
Proving the Quadratic nth term rule
I take it that we all know that halving the second difference of a quadratic sequence will give you the coefficient of n2?! However, did you know that there is a nice proof to show this? Start with the general form of a quadratic (an² + bn + c) and simply generate the first few terms with n=1, n=2 etc.
We all know the Fibonacci sequences and the sequences of square numbers etc, but what about:
- 1 + ½ + ¼ + 1/8… This is the series of terms of the form 1/2n and has some scope for discussion and pictorial demonstrations of its sum
- 1 + ½ + 1/3 + ¼ + 1/5… This is the harmonic series. Its sum is, perhaps surprisingly, infinity. Links here with music, integration and the block stacking problem
- 1 + 2 + 4 + 8 + 16… This is the series of term of the form 2n. The famous rice on a chessboard problem is a good example of exponential growth
- Pascal’s Triangle is a good source of many interesting sequences, including powers of 11 and the triangular numbers
The Fibonacci sequence is well worth exploring with a spreadsheet. Students get a chance to practice their Excel skills and it lets them test predictions and variations efficiently. The golden ratio can be found quickly using spreadsheets. Also, what if you varied the first numbers from 1, 1 to something else?
Littov’s Chain is another that lends itself well do spreadsheet exploration. This idea is from the excellent Don Steward Median blog. It goes like this:
- Choose any 2 numbers
- The next number is the mean average of these two
- The next number is the mean average of the previous two. Repeat
What is the limit of this sequence? What is you repeat the above but with three numbers and by taking the average of those 3?
Finally, I will leave you with a nice question from the 2019 WJEC GCE Unit 3 Exam:
- The 3rd, 19th and 67th terms of an arithmetic sequence form a geometric sequence.
Given that the arithmetic sequence is increasing and that the first term is 3, find the
common difference of the arithmetic sequence.