If `a_(1)=5 and a_(n)=1+sqrt(a_(n-1)),` find `a_(3)`.

Options

A. 2.623
B. 2.635
C. 2.673
D. 2.799

Explanation

To find \( a_3 \) given that \( a_1 = 5 \) and \( a_n = 1 + \sqrt{a_{n-1}} \), we will follow these steps:

### Step 1: Find \( a_2 \)
Using the recursive formula:
\[ a_2 = 1 + \sqrt{a_1} \]
Substituting the value of \( a_1 \):
\[ a_2 = 1 + \sqrt{5} \]
Calculating \( \sqrt{5} \):
\[ \sqrt{5} \approx 2.236 \]
Now substituting this value back:
\[ a_2 \approx 1 + 2.236 = 3.236 \]

### Step 2: Find \( a_3 \)
Now, we will use the value of \( a_2 \) to find \( a_3 \):
\[ a_3 = 1 + \sqrt{a_2} \]
Substituting the value of \( a_2 \):
\[ a_3 = 1 + \sqrt{3.236} \]
Calculating \( \sqrt{3.236} \):
\[ \sqrt{3.236} \approx 1.797 \]
Now substituting this value back:
\[ a_3 \approx 1 + 1.797 = 2.797 \]

### Final Answer
Thus, the value of \( a_3 \) is approximately:
\[ \boxed{2.797} \]