Question

8.1 defining and using sequences and series (continued)
1 exploration: writing rules for sequences (continued)
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b.
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communicate your answer

1. how can you write a rule for the nth term of a sequence?
2. what do you notice about the relationship between the terms in (a) an arithmetic sequence and (b) a geometric sequence? justify your answers.

Explanation:

To write the rule for the \( n \)-th term of a sequence, first, identify if the sequence is arithmetic, geometric, or neither. For an arithmetic sequence, the \( n \)-th term is given by \( a_n = a_1 + (n - 1)d \), where \( a_1 \) is the first term and \( d \) is the common difference. For a geometric sequence, the \( n \)-th term is \( a_n = a_1 \cdot r^{n - 1} \), where \( r \) is the common ratio. If the sequence is neither, analyze the pattern (e.g., quadratic, exponential) by examining the differences or ratios of terms and derive a formula based on the pattern observed.

Answer:

To write the \( n \)-th term rule: 1. Identify the sequence type (arithmetic, geometric, or other). 2. For arithmetic: \( a_n = a_1 + (n - 1)d \) ( \( a_1 \): first term, \( d \): common difference). 3. For geometric: \( a_n = a_1 \cdot r^{n - 1} \) ( \( r \): common ratio). 4. For other sequences, analyze term patterns (e.g., differences, ratios) to derive a formula.