To complete the table for the first 10 terms of the sequence defined by [tex]\(a_1 = 9\)[/tex] and [tex]\(a_{n+1} = \sqrt{2 + \sqrt{a_n}}\)[/tex], we need to calculate each term:
1. [tex]\(a_1 = 9\)[/tex]
2. To find [tex]\(a_2\)[/tex]:
[tex]\[
a_2 = \sqrt{2 + \sqrt{a_1}} = \sqrt{2 + \sqrt{9}} = \sqrt{2 + 3} = \sqrt{5} \approx 2.2361
\][/tex]
3. To find [tex]\(a_3\)[/tex]:
[tex]\[
a_3 = \sqrt{2 + \sqrt{a_2}} = \sqrt{2 + \sqrt{2.2361}} \approx \sqrt{2 + 1.4954} \approx \sqrt{3.4954} \approx 1.8696
\][/tex]
4. To find [tex]\(a_4\)[/tex]:
[tex]\[
a_4 = \sqrt{2 + \sqrt{a_3}} = \sqrt{2 + \sqrt{1.8696}} \approx \sqrt{2 + 1.3671} \approx \sqrt{3.3671} \approx 1.8350
\][/tex]
5. To find [tex]\(a_5\)[/tex]:
[tex]\[
a_5 = \sqrt{2 + \sqrt{a_4}} = \sqrt{2 + \sqrt{1.8350}} \approx \sqrt{2 + 1.3548} \approx \sqrt{3.3548} \approx 1.8316
\][/tex]
6. To find [tex]\(a_6\)[/tex]:
[tex]\[
a_6 = \sqrt{2 + \sqrt{a_5}} = \sqrt{2 + \sqrt{1.8316}} \approx \sqrt{2 + 1.3535} \approx \sqrt{3.3535} \approx 1.8312
\][/tex]
7. To find [tex]\(a_7\)[/tex]:
[tex]\[
a_7 = \sqrt{2 + \sqrt{a_6}} = \sqrt{2 + \sqrt{1.8312}} \approx \sqrt{2 + 1.3533} \approx \sqrt{3.3533} \approx 1.8312
\][/tex]
8. To find [tex]\(a_8\)[/tex]:
[tex]\[
a_8 = \sqrt{2 + \sqrt{a_7}} = \sqrt{2 + \sqrt{1.8312}} \approx \sqrt{2 + 1.3533} \approx \sqrt{3.3533} \approx 1.8312
\][/tex]
9. To find [tex]\(a_9\)[/tex]:
[tex]\[
a_9 = \sqrt{2 + \sqrt{a_8}} = \sqrt{2 + \sqrt{1.8312}} \approx \sqrt{2 + 1.3533} \approx \sqrt{3.3533} \approx 1.8312
\][/tex]
10. To find [tex]\(a_{10}\)[/tex]:
[tex]\[
a_{10} = \sqrt{2 + \sqrt{a_9}} = \sqrt{2 + \sqrt{1.8312}} \approx \sqrt{2 + 1.3533} \approx \sqrt{3.3533} \approx 1.8312
\][/tex]
Thus, the completed table is:
[tex]\[
\begin{array}{|c|c|}
\hline
n & a_n\\
\hline
1 & 9 \\
2 & 2.2361 \\
3 & 1.8696 \\
4 & 1.8350 \\
5 & 1.8316 \\
6 & 1.8312 \\
7 & 1.8312 \\
8 & 1.8312 \\
9 & 1.8312 \\
10 & 1.8312 \\
\hline
\end{array}
\][/tex]
