Math Problem Statement
an= 3^n + 7 / 5^n - 33
Solution
To analyze the sequence defined by , we can look at its behavior as approaches infinity.
Limit Analysis
-
Numerator: As increases, grows exponentially, while 7 becomes negligible in comparison.
- Thus, .
-
Denominator: Similarly, grows much faster than 33.
- Thus, .
Limit Calculation
Now, we find the limit of as :
Since , this limit approaches 0.
Conclusion
- The sequence converges to 0 as approaches infinity.
If you have specific questions about this sequence or want to explore further, feel free to ask!
Related Questions
- How does the convergence behavior of this sequence compare to other sequences?
- Can you derive the first few terms of the sequence ?
- What happens to for negative values of ?
- How would the sequence change if the numerator was modified to ?
- What is the significance of the terms and in this context?
Tip
When analyzing sequences, consider the dominant terms in both the numerator and denominator to determine limits effectively.
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Math Problem Analysis
Mathematical Concepts
Sequences
Limits
Exponential Functions
Formulas
a_n = (3^n + 7) / (5^n - 33)
lim (n -> ∞) (3^n / 5^n) = (3/5)^n
Theorems
Limit Theorem
Suitable Grade Level
Grades 10-12
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