From GCSE Sequences to Calculus

As part of the recent mathsconf39 I ran a session on using structure to promote algebraic thinking, and as part of that I looked at using a couple of different structures that can be used to help with the teaching of sequences. This got me reflecting on a link that I taught and highlighted to my Level 2 Further Maths and A-Level students about the links between the sequences they learn at GCSE/pre-GCSE and the calculus with polynomial functions that they first learned at Level 2 Further Maths or A-Level.

Part of the first work that pupils do with sequences is learn about linear sequences, and their defining characteristic being that of constant difference. Later they learn about quadratic sequences, and their defining characteristic being constant second difference. I always highlight that this means that the differences in the terms for a quadratic sequence form their own linear sequence. This establishes a progression from quadratic to linear to constant, or, in reverse, constant to linear to quadratic. I then ask them to predict what type of sequence would have a quadratic difference – which pupils quickly identify as cubic and can then extrapolate further from there.

This of course perfectly mirrors the progression that different polynomials go through when they are differentiated or integrated. Cubics differentiate to quadratics, which differentiate to linears, which differentiate to constants. So, when I first introduce differentiation to pupils, I ask them about where they have seen this progression before. Someone almost invariably mentions sequences, and if they don’t, I might start by writing out a quadratic sequence to prompt them.

A great question to ask students then is, of course, why? Why this connection between sequences and differentiation? This allows me to reinforce the idea of differentiation as about the rate of change, as when we are identifying the type of sequence we are examining the change between the terms.

It is worth then examining a particular sequence and its nth term, and the links to differentiation. I will typically pick one that has a coefficient of n², such as the sequence 3n² + 4n – 1:

n = 1		= 2		= 3		= 4
6		19		38		63
	+13		+19		+25
		+6		+6

 

If we differentiate the nth term of the sequence, we of course get 6n + 4, whereas the nth term of the linear 1^st differences is 6n + 7. At first this appears to be a discrepancy, until you see that the 1^st differences are in the 1.5^th, 2.5^th and 3.5^th position. We can account for this by shifting each value back by 3 (half of 6):

n = 1		= 2		= 3		= 4
6		19		38		63
10		16		22		28
	+6		+6		+6

 

Or by substituting n = 1.5 and T_n = 13 into 6n + a = T_n:

6 × 1.5 + a = 13

9 + a = 13

a = 4

Either way the nth term of the first differences can be shown to actually be 6n + 4, which is precisely what we get when we differentiate the nth term.

This provides a nice way of interleaving other algebraic manipulation and structure into the study of calculus, as well as reinforcing the idea that differentiation and sequences are two sides of the same mathematical idea – sequences being an introduction the study of changes and growth of discrete functions whilst differentiation (and integration) being the study of changes and growth of continuous functions. It also just adds to the impression for pupils that the landscape of mathematics is connected in unexpected but beautiful ways.