Problem 51 - Algebra and Trigonometry with Analytic Geometry

Question

Consider the sequence defined recursively by \(a_{1}=5\), \(a_{k+1}=\sqrt{a_{k}}\) for \(k \geq 1\). Describe what happens to the terms of the sequence as \(k\) increases.

Step-by-Step Solution

Answer

The terms of the sequence converge to 1 as \( k \to \infty \).

Step 1: Understanding the Sequence Definition

The sequence is defined with the first term \( a_1 = 5 \). Each subsequent term is the square root of the preceding term. So, \( a_{k+1} = \sqrt{a_k} \).

Step 2: Calculate Initial Terms

Let's calculate the first few terms. Starting with \( a_1 = 5 \), we find \( a_2 = \sqrt{5} \), \( a_3 = \sqrt{\sqrt{5}}\), and \( a_4 = \sqrt{\sqrt{\sqrt{5}}} \).

Step 3: Recognizing a Pattern

Notice that each term becomes the square root of the previous term. By repeated extraction of the square root, each term gets smaller and approaches 1.

Step 4: Identifying the Limit

As \( k \) increases, since the square root of a number slightly larger than 1 is a number closer to 1, the terms of the sequence converge to 1. This is because repeatedly taking square roots of a number greater than 1 gradually pulls it towards 1.

Step 5: Conclusion

Therefore, the terms of the sequence \( a_k \) converge to 1 as \( k \) becomes very large.

Key Concepts

Sequence Convergence

A recurring topic in the study of recursive sequences is sequence convergence. A sequence is said to converge when its terms approach a specific value as the number of terms increases. This specific value is known as the limit of the sequence. In the case of the sequence defined by \(a_{1}=5\) and \(a_{k+1}=\sqrt{a_{k}}\), convergence happens as we observe the terms becoming progressively closer to the number 1.

As we progress through more terms, like \(a_2=\sqrt{5}\) and \(a_3=\sqrt{\sqrt{5}}\), each term is closer to 1 than the last. The reason is simple: when repeatedly taking the square root of a number greater than 1, the results become smaller and move closer to 1. This is fundamental to understanding why the sequence converges to 1. The convergence provides crucial insights into the long-term behavior of sequences and is particularly useful in mathematical analysis of sequences.

Limits of Sequences

The limit of a sequence is the value that the terms of the sequence approach as the index (often \(k\) in mathematical notation) tends to infinity. The limit tells us about the eventual behavior of the sequence, and it plays a crucial role in calculus and analytical mathematics.

For the sequence in question, the limit is 1. As we calculated earlier from the initial terms, \(a_1=5\), \(a_2=\sqrt{5}\), and so on, each term approaches closer to 1.

• When a sequence converges to a specific value, it is said to be convergent, which indicates stable behavior in the terms as \(k\rightarrow \infty\).
• Understanding limits helps us predict the long-term behavior without computing all terms indefinitely.

In practical applications, finding the limit of a sequence can help solve complex problems in various fields like physics, engineering, and economics, where predicting steady-state outcomes is essential.

Square Roots

Square roots can deeply impact a sequence's behavior, especially in recursively defined sequences like this one. The square root operation is a unique mathematical function because it outputs a number that, when multiplied by itself, returns the original number. In simpler terms, \(\sqrt{x} = y\) if \(y \times y = x\).

For our sequence, taking square roots continuously decreases terms greater than 1 and increases terms less than 1, pulling both cases towards a balancing point of 1. Here, since \(a_1 = 5\) is greater than 1, each subsequent term is decreased, nudging the sequence closer to its limit.

• Square roots always yield non-negative results for non-negative inputs, maintaining the non-negativity of the sequence.
• The recursive nature of taking successive square roots ensures the terms are shrinking, illustrating the power of iterative processes in reaching convergence.

In broader mathematics, understanding square roots is pivotal not just in analyzing sequences, but also in solving quadratic equations, optimization problems, and understanding waveforms in engineering.