Let $a_{1}, a_{2}, a_{3}, \ldots, a_{n}, \ldots$ be a geometric sequence. Find $S_{10}$ and $S_{\infty}$. \[ a_{1}=400, r=0.9 \] \[ S_{10}= \] (Type an integer or decimal rounded to two decimal places as needed.)

Step 1 ：Let's denote the first term of the geometric sequence as \(a_1 = 400\), the common ratio as \(r = 0.9\), and the number of terms as \(n = 10\) for \(S_{10}\).

Step 2 ：The formula for the sum of the first n terms of a geometric sequence is given by: \[S_n = a_1 * \frac{1 - r^n}{1 - r}\]

Step 3 ：Substituting the given values into the formula, we get \(S_{10} = 400 * \frac{1 - 0.9^{10}}{1 - 0.9}\)

Step 4 ：Calculating the above expression, we find that \(S_{10} = 2605.2862396000005\)

Step 5 ：The formula for the sum to infinity of a geometric sequence is given by: \[S_{\infty} = \frac{a_1}{1 - r}\] This formula is valid only if \(|r| < 1\), which is true in this case since \(r = 0.9\).

Step 6 ：Substituting the given values into the formula, we get \(S_{\infty} = \frac{400}{1 - 0.9}\)

Step 7 ：Calculating the above expression, we find that \(S_{\infty} = 4000.000000000001\)

Step 8 ：Final Answer: The sum of the first 10 terms of the geometric sequence, \(S_{10}\), is approximately \(\boxed{2605.29}\) and the sum to infinity of the geometric sequence, \(S_{\infty}\), is \(\boxed{4000}\).