To find the 10th term of the sequence [tex]\( \sqrt{2}, \sqrt{8}, \sqrt{18}, \ldots \)[/tex], we need to identify the pattern in the terms.
Let's look at the given terms closely:
1. The first term [tex]\( \sqrt{2} \)[/tex] is [tex]\( \sqrt{2 \cdot 1^2} \)[/tex].
2. The second term [tex]\( \sqrt{8} \)[/tex] is [tex]\( \sqrt{2 \cdot 4} = \sqrt{2 \cdot 2^2} \)[/tex].
3. The third term [tex]\( \sqrt{18} \)[/tex] is [tex]\( \sqrt{2 \cdot 9} = \sqrt{2 \cdot 3^2} \)[/tex].
From these observations, we can generalize that the [tex]\( n \)[/tex]-th term of the sequence can be written as:
[tex]\[ a_n = \sqrt{2 \cdot n^2} \][/tex]
This simplifies to:
[tex]\[ a_n = n \cdot \sqrt{2} \][/tex]
Now we want to find the 10th term of the sequence, which means we set [tex]\( n = 10 \)[/tex]:
[tex]\[ a_{10} = 10 \cdot \sqrt{2} \][/tex]
Evaluating this, we get:
[tex]\[ a_{10} \approx 10 \cdot 1.4142135623730951 \][/tex]
[tex]\[ a_{10} \approx 14.142135623730951 \][/tex]
Therefore, the 10th term of the sequence is approximately [tex]\( 14.142135623730951 \)[/tex].
