Math Problem Statement

an= 3^n + 7 / 5^n - 33

Solution

To analyze the sequence defined by an=3n+75n33a_n = \frac{3^n + 7}{5^n - 33}, we can look at its behavior as nn approaches infinity.

Limit Analysis

  1. Numerator: As nn increases, 3n3^n grows exponentially, while 7 becomes negligible in comparison.

    • Thus, 3n+73n3^n + 7 \sim 3^n.
  2. Denominator: Similarly, 5n5^n grows much faster than 33.

    • Thus, 5n335n5^n - 33 \sim 5^n.

Limit Calculation

Now, we find the limit of ana_n as nn \to \infty:

limnan=limn3n+75n33=limn3n5n=limn(35)n\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{3^n + 7}{5^n - 33} = \lim_{n \to \infty} \frac{3^n}{5^n} = \lim_{n \to \infty} \left( \frac{3}{5} \right)^n

Since 35<1\frac{3}{5} < 1, this limit approaches 0.

Conclusion

  • The sequence ana_n converges to 0 as nn approaches infinity.

If you have specific questions about this sequence or want to explore further, feel free to ask!

Related Questions

  1. How does the convergence behavior of this sequence compare to other sequences?
  2. Can you derive the first few terms of the sequence ana_n?
  3. What happens to ana_n for negative values of nn?
  4. How would the sequence change if the numerator was modified to 2n+72^n + 7?
  5. What is the significance of the terms 3n3^n and 5n5^n 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