# Indexing Sequences: Do We Start at 0 or 1?

• B
• Mr Davis 97
In summary, the conversation discusses the issue of indexing periodic sequences. The question is whether to designate the first term as 0 or 1, and how this affects finding the nth term using n mod 4. It is mentioned that there are four different ways to index the elements of the sequence and the vocabulary set, and two of them are presented. It is also noted that starting at 0 is usually the simplest way for periodic sequences, but some computer languages only allow array indexing to start at 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?

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)}##

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

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.

## What is the Nth term of a periodic sequence?

The Nth term of a periodic sequence is the general term of a sequence that follows a repeating pattern. It is the formula used to find the value of any term in the sequence using the term number (N) as the input.

## How do you find the Nth term of a periodic sequence?

To find the Nth term of a periodic sequence, you need to first identify the pattern of the sequence. Then, use the pattern to create a general formula for finding any term in the sequence. Substitute the term number (N) into the formula to find the value of the desired term.

## Can a periodic sequence have more than one Nth term?

No, a periodic sequence can only have one Nth term. This is because the Nth term represents the general formula for finding any term in the sequence. If there were multiple Nth terms, it would mean that the sequence has multiple patterns or rules, which would make it non-periodic.

## What is the difference between a periodic sequence and a non-periodic sequence?

A periodic sequence follows a repeating pattern, meaning that the same set of numbers or variables will occur in the same order after a certain number of terms. In contrast, a non-periodic sequence does not have a repeating pattern and the numbers or variables in the sequence do not follow a specific rule or pattern.

## Can the Nth term of a periodic sequence be negative?

Yes, the Nth term of a periodic sequence can be negative. This will depend on the specific pattern and rule of the sequence. Some periodic sequences may have negative values in their Nth term, while others may not.

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