a_(n+1) = sqrt(2+a_n), a_1 = sqrt(2), prove that the serie is decreasing

We are given the recursive sequence defined by:

\[
a_{n+1} = \sqrt{2 + a_n}, \quad a_1 = \sqrt{2}.
\]

We are tasked with proving that the sequence is **decreasing**.

### Step 1: Checking if the sequence is bounded

We will first check whether the sequence has an upper bound. Notice that:

\[
a_1 = \sqrt{2}.
\]

Since \( a_{n+1} = \sqrt{2 + a_n} \), clearly \( a_{n+1} \geq \sqrt{2} \) for all \( n \), meaning the sequence cannot go below \( \sqrt{2} \). Hence, the sequence is bounded below by \( \sqrt{2} \).

### Step 2: Prove that the sequence is decreasing

Now, to prove that the sequence is decreasing, we need to show that:

\[
a_{n+1} \leq a_n \quad \text{for all } n.
\]

This implies showing that:

\[
\sqrt{2 + a_n} \leq a_n \quad \text{for all } n.
\]

#### Step 2.1: Squaring both sides

To eliminate the square root, we can square both sides:

\[
(\sqrt{2 + a_n})^2 \leq a_n^2.
\]

This simplifies to:

\[
2 + a_n \leq a_n^2.
\]

Rearranging terms gives:

\[
a_n^2 - a_n - 2 \geq 0.
\]

Factoring the quadratic expression:

\[
(a_n - 2)(a_n + 1) \geq 0.
\]

#### Step 2.2: Analyzing the inequality

For \( (a_n - 2)(a_n + 1) \geq 0 \), there are two possibilities:
- \( a_n \geq 2 \), or
- \( a_n \leq -1 \).

Since we know from the recursive relation that \( a_n \geq \sqrt{2} \approx 1.414 \), the condition \( a_n \geq 2 \) never holds for our sequence, which starts below 2. Thus, for the inequality to hold, the sequence cannot be decreasing for values less than 2.

### Conclusion:

The sequence is **not decreasing** since the inequality \( a_{n+1} \leq a_n \) is not satisfied in the range of values \( 1.414 \leq a_n \leq 2 \).

Would you like to review any part of this reasoning? Here are five related questions to deepen your understanding:

1. How do you rigorously show that a sequence is bounded?
2. Can a sequence converge if it is not monotonic?
3. What additional conditions are needed to prove that a sequence converges?
4. How does the initial value of \( a_1 \) affect the behavior of the sequence?
5. Can you use an alternate approach, like induction, to test if this sequence is decreasing?

**Tip:** When working with sequences, always check if they are bounded and monotonic to infer convergence.

a_(n+1) = sqrt(2 + a_n), a_1 = sqrt(2), prove that the series is decreasing

This solution explores the properties of the recursive sequence a_(n+1) = sqrt(2 + a_n), starting with a_1 = sqrt(2). The problem involves checking for monotonicity and boundedness using inequalities. Suitable for students in Grades 11-12.

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