Math Problem Statement
Recursive formula for the sequence represented by the explicit formula a n=1/5(7)n-1
Solution
To find the recursive formula for the sequence represented by the explicit formula:
we observe the relationship between consecutive terms.
Step-by-step process:
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Explicit formula:
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Compute the ratio of consecutive terms: This shows that each term is 7 times the previous term.
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Recursive formula: Using the ratio between consecutive terms, we can write the recursive formula as: with the initial condition:
Final Recursive Formula:
Would you like a detailed explanation of how recursive formulas are derived or their applications?
Related Questions:
- How do explicit and recursive formulas differ in practice?
- What is the general method to convert between explicit and recursive formulas?
- How can we determine the -th term directly using a recursive formula?
- Can a sequence have more than one recursive formula representation?
- What happens if we change the base of the exponential term in the sequence?
Tip:
To verify a recursive formula, test it with small values of to ensure it correctly generates the sequence from the explicit formula.
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Math Problem Analysis
Mathematical Concepts
Sequences and Series
Recursive Formulas
Exponential Functions
Formulas
Explicit formula: a_n = (1/5)(7)^(n-1)
Recursive formula: a_n = 7 * a_(n-1)
Theorems
Properties of Exponential Functions
Recursive Sequence Relations
Suitable Grade Level
Grades 9-12
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