- Difference between Recursive Formula and Explicit Formula:
A recursive formula defines each term in a sequence based on the previous term(s). This means you need the previous values to calculate the next term. In contrast, an explicit formula provides a direct way to compute the value of any term in the sequence without needing to reference prior terms. You can find the value of a term based solely on its position in the sequence.
- Determine if a Sequence is Recursive or Explicit:
To determine whether a sequence is recursive or explicit, you would look at how the terms are defined:
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If a sequence is defined in such a way that each term uses the value(s) of previous term(s), it is recursive. For example, if a sequence is defined as \( a_n = a_{n-1} + 2 \) with a base case, like \( a_1 = 1 \), this is recursive.
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If a sequence provides a formula that directly calculates the term based on its position (like \( a_n = 3n + 1 \)), it is explicit.
You can analyze the given sequence based on these definitions to categorize it properly. If you have a specific sequence in mind, feel free to share it for further analysis!
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