To determine the 10th term of the sequence [tex]\(\sqrt{8}, \sqrt{18}, \sqrt{32}, \ldots\)[/tex], let's analyze the pattern.
1. The first term is [tex]\(\sqrt{8}\)[/tex]:
[tex]\[
\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}
\][/tex]
2. The second term is [tex]\(\sqrt{18}\)[/tex]:
[tex]\[
\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}
\][/tex]
3. The third term is [tex]\(\sqrt{32}\)[/tex]:
[tex]\[
\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}
\][/tex]
From these calculations, we observe that each term follows the general form:
[tex]\[
n\sqrt{2}
\][/tex]
where [tex]\( n \)[/tex] is the natural number (1, 2, 3, ...).
For the 10th term, [tex]\( n = 11 \)[/tex] (since it follows [tex]\((n+1)\sqrt{2}\)[/tex] form). Therefore, the 10th term is:
[tex]\[
11\sqrt{2}
\][/tex]
Next, we convert [tex]\( 11\sqrt{2} \)[/tex] back to the form that matches the options provided. Doing this, we square both sides:
[tex]\[
(11\sqrt{2})^2 = 11^2 \times 2 = 121 \times 2 = 242
\][/tex]
Thus, the 10th term is [tex]\(\sqrt{242}\)[/tex].
Therefore, the correct answer is:
[tex]\[
\textbf{b) } \sqrt{242}
\][/tex]
