12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = (2ⁿ + 5ⁿ⁺¹) / 5ⁿ

Verified step by step guidance

1. Identify the given sequence: \(a_n = \frac{2^n + 5^{n+1}}{5^n}\).
2. Rewrite the sequence by separating the terms in the numerator over the denominator: \(a_n = \frac{2^n}{5^n} + \frac{5^{n+1}}{5^n}\).
3. Simplify each term: \(\frac{2^n}{5^n} = \left(\frac{2}{5}\right)^n\) and \(\frac{5^{n+1}}{5^n} = 5^{(n+1)-n} = 5\).
4. Express the sequence as \(a_n = \left(\frac{2}{5}\right)^n + 5\).
5. Analyze the limit as \(n\) approaches infinity: since \(\left(\frac{2}{5}\right)^n\) tends to 0 (because \(\frac{2}{5} < 1\)), the limit of \(a_n\) is the limit of \$0 + 5$, which is 5.

Key Concepts

Limits of Sequences

The limit of a sequence describes the value that the terms of the sequence approach as the index n becomes very large. If the terms get arbitrarily close to a specific number, the sequence converges to that limit; otherwise, it diverges.

Properties of Exponential Functions

Exponential functions with different bases grow at different rates. When comparing terms like 2ⁿ and 5ⁿ, the term with the larger base dominates as n increases, which helps simplify the limit by focusing on the dominant term.

Algebraic Manipulation of Sequences

Simplifying sequences often involves factoring and dividing numerator and denominator by the highest power term to reveal dominant behavior. This technique helps in evaluating limits by reducing complex expressions to simpler forms.