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Sagot :
To complete the table for the first 10 terms of the sequence where [tex]\( a_1 = 9 \)[/tex] and [tex]\( a_{n+1} = \sqrt{2 + \sqrt{a_n}} \)[/tex], we need to follow the recursive formula. Let's go through each step:
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]\[ a_3 = \sqrt{2 + \sqrt{a_2}} = \sqrt{2 + \sqrt{2.2361}} \approx \sqrt{2 + 1.4954} = \sqrt{3.4954} \approx 1.8696 \][/tex]
4. To find [tex]\( a_4 \)[/tex]:
[tex]\[ a_4 = \sqrt{2 + \sqrt{a_3}} = \sqrt{2 + \sqrt{1.8696}} \approx \sqrt{2 + 1.3670} = \sqrt{3.3670} \approx 1.8350 \][/tex]
5. To find [tex]\( a_5 \)[/tex]:
[tex]\[ a_5 = \sqrt{2 + \sqrt{a_4}} = \sqrt{2 + \sqrt{1.8350}} \approx \sqrt{2 + 1.3546} = \sqrt{3.3546} \approx 1.8316 \][/tex]
6. To find [tex]\( a_6 \)[/tex]:
[tex]\[ a_6 = \sqrt{2 + \sqrt{a_5}} = \sqrt{2 + \sqrt{1.8316}} \approx \sqrt{2 + 1.3534} = \sqrt{3.3534} \approx 1.8312 \][/tex]
7. To find [tex]\( a_7 \)[/tex]:
[tex]\[ a_7 = \sqrt{2 + \sqrt{a_6}} = \sqrt{2 + \sqrt{1.8312}} \approx \sqrt{2 + 1.3532} = \sqrt{3.3532} \approx 1.8312 \][/tex]
8. To find [tex]\( a_8 \)[/tex]:
[tex]\[ a_8 = \sqrt{2 + \sqrt{a_7}} = \sqrt{2 + \sqrt{1.8312}} \approx \sqrt{2 + 1.3532} = \sqrt{3.3532} \approx 1.8312 \][/tex]
9. To find [tex]\( a_9 \)[/tex]:
[tex]\[ a_9 = \sqrt{2 + \sqrt{a_8}} = \sqrt{2 + \sqrt{1.8312}} \approx \sqrt{2 + 1.3532} = \sqrt{3.3532} \approx 1.8312 \][/tex]
10. To find [tex]\( a_{10} \)[/tex]:
[tex]\[ a_{10} = \sqrt{2 + \sqrt{a_9}} = \sqrt{2 + \sqrt{1.8312}} \approx \sqrt{2 + 1.3532} = \sqrt{3.3532} \approx 1.8312 \][/tex]
So, 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]
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]\[ a_3 = \sqrt{2 + \sqrt{a_2}} = \sqrt{2 + \sqrt{2.2361}} \approx \sqrt{2 + 1.4954} = \sqrt{3.4954} \approx 1.8696 \][/tex]
4. To find [tex]\( a_4 \)[/tex]:
[tex]\[ a_4 = \sqrt{2 + \sqrt{a_3}} = \sqrt{2 + \sqrt{1.8696}} \approx \sqrt{2 + 1.3670} = \sqrt{3.3670} \approx 1.8350 \][/tex]
5. To find [tex]\( a_5 \)[/tex]:
[tex]\[ a_5 = \sqrt{2 + \sqrt{a_4}} = \sqrt{2 + \sqrt{1.8350}} \approx \sqrt{2 + 1.3546} = \sqrt{3.3546} \approx 1.8316 \][/tex]
6. To find [tex]\( a_6 \)[/tex]:
[tex]\[ a_6 = \sqrt{2 + \sqrt{a_5}} = \sqrt{2 + \sqrt{1.8316}} \approx \sqrt{2 + 1.3534} = \sqrt{3.3534} \approx 1.8312 \][/tex]
7. To find [tex]\( a_7 \)[/tex]:
[tex]\[ a_7 = \sqrt{2 + \sqrt{a_6}} = \sqrt{2 + \sqrt{1.8312}} \approx \sqrt{2 + 1.3532} = \sqrt{3.3532} \approx 1.8312 \][/tex]
8. To find [tex]\( a_8 \)[/tex]:
[tex]\[ a_8 = \sqrt{2 + \sqrt{a_7}} = \sqrt{2 + \sqrt{1.8312}} \approx \sqrt{2 + 1.3532} = \sqrt{3.3532} \approx 1.8312 \][/tex]
9. To find [tex]\( a_9 \)[/tex]:
[tex]\[ a_9 = \sqrt{2 + \sqrt{a_8}} = \sqrt{2 + \sqrt{1.8312}} \approx \sqrt{2 + 1.3532} = \sqrt{3.3532} \approx 1.8312 \][/tex]
10. To find [tex]\( a_{10} \)[/tex]:
[tex]\[ a_{10} = \sqrt{2 + \sqrt{a_9}} = \sqrt{2 + \sqrt{1.8312}} \approx \sqrt{2 + 1.3532} = \sqrt{3.3532} \approx 1.8312 \][/tex]
So, 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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