To find the common ratio of a geometric sequence, we need to use the ratio of any term in the sequence to its preceding term. For the given geometric sequence [tex]\((\sqrt{2}-1),(3-2\sqrt{2}), \ldots\)[/tex], let's denote the first term as [tex]\(a_1\)[/tex] and the second term as [tex]\(a_2\)[/tex].
1. The first term of the sequence is:
[tex]\[
a_1 = \sqrt{2} - 1
\][/tex]
2. The second term of the sequence is:
[tex]\[
a_2 = 3 - 2\sqrt{2}
\][/tex]
3. The common ratio [tex]\(r\)[/tex] of a geometric sequence is found by dividing the second term by the first term:
[tex]\[
r = \frac{a_2}{a_1}
\][/tex]
4. Substituting the given terms into the formula:
[tex]\[
r = \frac{3 - 2\sqrt{2}}{\sqrt{2} - 1}
\][/tex]
5. Given the calculated value, the common ratio [tex]\(r\)[/tex] is found to be approximately:
[tex]\[
r = 0.4142135623730945
\][/tex]
So, the common ratio of the geometric sequence is approximately [tex]\(0.4142135623730945\)[/tex].
