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Complete the table below for the first 10 terms of the sequence whose terms are given by [tex]\( a_1 = 9 \)[/tex] and [tex]\( a_{n+1} = \sqrt{2 + \sqrt{a_n}} \)[/tex].
(Round to four decimal places as needed.)

[tex]\[
\begin{tabular}{|c|c|}
\hline
n & u(n) = a_n \\
\hline
1 & 9 \\
2 & \square \\
3 & \square \\
4 & \square \\
5 & \square \\
6 & \square \\
7 & \square \\
8 & \square \\
9 & \square \\
10 & \square \\
\hline
\end{tabular}
\][/tex]

Sagot :

GinnyAnsweridn
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]
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