A sequence is an ordered list of numbers, called terms, that follow a specific pattern or rule. Each number in the sequence is referred to as a term, and the position of a term in the sequence is its term number.
There are many types of sequences, but the most common ones are:
- Arithmetic sequences – The difference between consecutive terms is constant. For example, the sequence 1, 3, 5, 7, 9… is an arithmetic sequence with a common difference of 2.
- Geometric sequences – The ratio between consecutive terms is constant. For example, the sequence 2, 4, 8, 16, 32… is a geometric sequence with a common ratio of 2.
- Quadratic sequences – The difference between consecutive terms grows linearly. In other words, the difference between consecutive terms is itself an arithmetic sequence. For example, the sequence 1, 4, 9, 16, 25… is a quadratic sequence, since the difference between consecutive terms is the arithmetic sequence 3, 5, 7, 9, …
Generating terms of a sequence can be done using either a term-to-term rule or a position-to-term rule.
- Term-to-term rule: Each term is generated based on the previous term(s) following a specific pattern. For arithmetic sequences, you add the common difference to the previous term to obtain the next term. Whereas, for geometric sequences, you multiply the previous term by the common ratio to find the next term. In general, term-to-term rules describe how to move from one term to another within the sequence.
- Position-to-term rule: The value of a term is determined based on its position (term number) within the sequence, using a formula. This allows you to find any term directly without calculating all preceding terms.
The Nᵗʰ term of a sequence is a formula or expression that helps us find the value of any term in the sequence without having to list all the previous terms. It allows us to calculate the value of a term at a specific position (n) in the sequence.
Each type of sequence has a unique Nᵗʰ term formula that reflects its specific pattern or rule. By identifying the correct Nᵗʰ term formula for a given sequence, we can easily calculate the value of any term in the sequence. Also, we can better understand the relationships between the terms.
‘n’ in the Nᵗʰ term formula represents the position of the term in the sequence. When n = 1, we are looking for the first term in the sequence, and when n = 2, we are looking for the second term in the sequence.