We are given a recursive sequence with:

$a_1 = 1 \quad \text{and} \quad a_{n+1} = \frac{1}{2} \left( a_n + \frac{4}{a_n} \right).$

We are asked to compute $a_5$.

Let's calculate $a_2, a_3, a_4,$ and $a_5$ step by step.

Step-by-Step Calculation:

1. For $a_2$: $a_2 = \frac{1}{2} \left( a_1 + \frac{4}{a_1} \right) = \frac{1}{2} \left( 1 + \frac{4}{1} \right) = \frac{1}{2} \times 5 = 2.5.$

2. For $a_3$: $a_3 = \frac{1}{2} \left( a_2 + \frac{4}{a_2} \right) = \frac{1}{2} \left( 2.5 + \frac{4}{2.5} \right) = \frac{1}{2} \left( 2.5 + 1.6 \right) = \frac{1}{2} \times 4.1 = 2.05.$

3. For $a_4$: $a_4 = \frac{1}{2} \left( a_3 + \frac{4}{a_3} \right) = \frac{1}{2} \left( 2.05 + \frac{4}{2.05} \right) = \frac{1}{2} \left( 2.05 + 1.9512 \right) = \frac{1}{2} \times 4.0012 = 2.0006.$

4. For $a_5$: $a_5 = \frac{1}{2} \left( a_4 + \frac{4}{a_4} \right) = \frac{1}{2} \left( 2.0006 + \frac{4}{2.0006} \right) = \frac{1}{2} \left( 2.0006 + 1.9994 \right) = \frac{1}{2} \times 4.0000 = 2.$

Conclusion:

Thus, $a_5 = 2$.

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. What happens to $a_n$ as $n$ becomes large?
2. Is there a closed-form expression for $a_n$?
3. Can we prove that $a_n$ converges, and if so, to what value?
4. How does this recursive sequence relate to the arithmetic-geometric mean?
5. How would the result change if the initial condition $a_1$ were different?

Tip: When working with recursive sequences, calculating a few terms manually can reveal patterns that may help predict long-term behavior.