Question

Describe the pattern, write the next term, and write a rule for the \(\boldsymbol{n}\) th term of the sequence. \(1,6,11,16, \ldots\)

Short answer

The next term in the sequence is 21, and the rule for the \(n^{th}\) term of the sequence is be \(a_n = 5n - 4\).

Step-by-step solution

Identifying the Pattern

Examine the given sequence to identify a pattern. The difference between two consecutive numbers can be determined by subtracting the former number from the latter. When this is done for the given sequence 6-1=5, 11-6=5, and 16-11=5, all the differences are the same and equal to 5. Thus, the sequence is arithmetic with a common difference of 5.

Predicting the Next Term

Given the common difference of an arithmetic sequence, the next term can easily be predicted by adding the common difference to the last term provided. In our case, the last term given is 16 and our common difference is 5, therefore 16 + 5 = 21. Therefore, the next term of the sequence is 21.

Formulating the Rule for the nth Term

The general rule for an arithmetic sequence can be given by: \(a_n = a_1 + (n-1) * d\), where \(a_n\) is the \(n^{th}\) term, \(a_1\) is the first term and \(d\) is the common difference. Substituting the values for our sequence into this formula, we get:\(a_n = 1 + (n-1)*5 = 5n - 4\)The rule for our sequence therefore is \(a_n = 5n - 4\).

Key concepts

Common Difference

In an arithmetic sequence, the common difference is a key element that defines the progression between consecutive terms. This difference is constant throughout the sequence, allowing for predictable advances from one term to the next. To find the common difference, simply subtract any term from the term that follows. This subtraction should always yield the same result in an arithmetic sequence. For example, in the sequence given, \(1, 6, 11, 16, \ldots\), subtracting 1 from 6 gives 5, 6 from 11 also gives 5, and so on. Therefore, the common difference of 5 indicates that each term increases by 5 compared to the previous term.

Recognizing the common difference not only helps in predicting future terms but also aids in understanding the structure of the sequence. Once identified, this difference allows for the construction of the entire sequence from the initial term onwards.

Sequence Patterns

Recognizing sequence patterns is an essential skill in mathematics which involves observing how numbers change from one term to the next. This helps not only in the identification of the sequence type but also in understanding how terms relate to each other. In an arithmetic sequence, the pattern is dictated by the common difference, where each term is increased or decreased by a fixed amount compared to the last.

For the provided sequence \(1, 6, 11, 16, \ldots\), the numbers consistently rise by 5. Noticeably, identifying this pattern helps predict the next number in the sequence without additional calculation. If the pattern breaks, it signals a different kind of sequence, highlighting the distinctiveness of arithmetic sequences and their predictable nature.

nth Term Rule

To find any term in an arithmetic sequence without listing all preceding terms, the nth term rule comes in handy. This mathematical formula expresses the general term \(a_n\) and can be applied universally to determine any term's position \(n\) in the sequence. The rule is expressed as:

• \(a_n = a_1 + (n-1) \cdot d\)

where:

• \(a_n\) is the nth term you want to find,
• \(a_1\) is the first term of the sequence,
• \(n\) is the term number,
• \(d\) is the common difference.

In our example, substituting 1 for \(a_1\), 5 for \(d\), we derive the nth term formula: \(a_n = 1 + (n-1) \cdot 5 = 5n - 4\). This formula enables you to compute any term directly, which is incredibly useful for large sequences or specific inquiries about distant terms.