Indexing Sequences: Do We Start at 0 or 1?

[Mr Davis 97]

Say we have a periodic sequencs, ABCDABCDABCDA... etc. We would normally call A term 1, B term 2, C term 3, etc. However, to find the nth term, do we need to designate A as term 0, B as term 1, etc? Since we would use n mod 4 to find the nth term, wouldn't this mean that 4, 8, 12, etc would have to correspond to 0 rather than 1, this showing that we have to start the sequence ordering from 0 rather than 1?
Wouldn't this be in contrast to other sequences such as geometric and arithmetic that start at 1 rather than 0?

 

[andrewkirk]

We can write it either way. It depends on how we index the four-element vocabulary set and also on how we index the elements of the sequence. Since either can start at 0 or 1, there are four different ways we can do this. Here are two of them:

Least compact method (both start at 1):
$V_1=`A',\ V_2=`B',\ V_3=`C', \ V_4=`D'$ then the sequence $s_n$, numbered starting at 1, has $s_n=V_{(1+(n-1)\mod 4)}$

Most compact method (both start at 0):
$V_0=`A',\ V_1=`B',\ V_2=`C', \ V_3=`D'$ then the sequence $s_n$, numbered starting at 0, has $s_n=V_{(n\mod 4)}$

 

[Mr Davis 97]

So although both ways of numbering are possible, starting at 0 is probably the simplest way for periodic sequences?

 

[andrewkirk]

Yes. This issue often comes up in programming computers. Deciding whether to index a vector starting with zero or one can significantly affect the verbosity of the code that refers to that vector. Usually, but not always, I have found it is better to start at 0. But not all computer languages allow array indexing to start with numbers other than 1.