{Use of Tech} Repeated square roots
Consider the sequence defined by
aₙ₊₁ = √(2 + aₙ),a₀ = √2, for n = 0, 1, 2, 3, …


a.Evaluate the first four terms of the sequence {aₙ}.
State the exact values first, and then the approximate values.

Verified step by step guidance

1. Start with the initial term given: \(a_0 = \sqrt{2}\). This is your starting point for the sequence.
2. Use the recursive formula \(a_{n+1} = \sqrt{2 + a_n}\) to find the next terms. For \(a_1\), substitute \(a_0\) into the formula: \(a_1 = \sqrt{2 + a_0}\).
3. Calculate \(a_2\) by substituting \(a_1\) into the formula: \(a_2 = \sqrt{2 + a_1}\). Keep the expression exact without approximating yet.
4. Similarly, find \(a_3\) by substituting \(a_2\) into the formula: \(a_3 = \sqrt{2 + a_2}\). Again, keep the exact form.
5. After writing down the exact expressions for \(a_0\), \(a_1\), \(a_2\), and \(a_3\), approximate each value using a calculator or technology to get decimal approximations.

Key Concepts

Recursive Sequences

A recursive sequence is defined by specifying one or more initial terms and a rule to find each subsequent term from the previous ones. In this problem, each term aₙ₊₁ depends on the previous term aₙ through the formula aₙ₊₁ = √(2 + aₙ). Understanding recursion helps in computing terms step-by-step.

Square Roots and Nested Radicals

Square roots involve finding a number which, when squared, returns the original value. Nested radicals, like √(2 + √(2 + ...)), appear in this sequence. Recognizing how to simplify or approximate these expressions is key to evaluating terms exactly and approximately.

Exact vs. Approximate Values

Exact values are expressed in precise mathematical form (e.g., √2), while approximate values are numerical estimates (e.g., 1.414). Calculating both helps understand the behavior of the sequence and provides insight into convergence or growth patterns.