In teaching high school mathematics where actions are typically focussed on an instant in time rather actions over a period of time, this is not always obvious.

When teaching recurring formula for an arithmetic sequence (I lack the ability to subscript properly in Blogger - subscript after T until a space is encountered) we usually discuss:

a, a+d, a+2d, a+3d,..., a+(n-1)d where n is a position in the sequence

Tn+1 = Tn + d or Tn = Tn-1 + d where T1 =a

It is not easy for students to recognise (especially in year 10) the n+1 or n-1 as describing position in a sequence.

eg for the sequence 1,3,5,7

T1=1, T2=3, T3=5, T4=7

when n=1; Tn=1, Tn+1 = 3, Tn-1 is undefined

when n=2; Tn=3, Tn+1 = 5, Tn-1 = 1

I had a think about it and next time I think will try teach it using an investigative introduction.

I will write on the board:

Describe in mathematical/algebriac terms my yearly salary based on "I get $10

000 in my first year and a pay rise of $1 000 each year thereafter."

And then determine the sequence for the first five terms and introduce recursive algorithms to give students more idea where this form of mathematics could lie within their internal schema. I would construct the sequence (10 000, 11 000, 12 000, ...) and have focus students on the concept of thinking of the $1000 as the difference(d) between terms and the $10 000 as the first term(T1 or 'a') - then introduce the following:

Tn = Tn-1 + 1000 ; n>1 and T1 = 10000

This I think would lead well to the introduction of Tn+1 = Tn + d and Tn = Tn-1 + d.

Then write on the board and investigate students answers for:

For the above problem find T100And then launch into Arithmetic Progressions (AP's) and the formula Tn = T1 + (n-1)d before completing a set of practice examples.