Answer :
To solve this problem, we first need to recognize the given parameters of the geometric sequence. You have the first term \(a_1\) and the common ratio \(r\) for the sequence. The general form for the \(n\)-th term of a geometric sequence is:
[tex]\[ a_n = a_1 \cdot r^{n-1} \][/tex]
For the given sequence:
[tex]\[ a_n = -256 \left(-\frac{1}{4}\right)^{n-1} \][/tex]
Here, \(a_1 = -256\) and \(r = -\frac{1}{4}\).
Next, we need to find the sum of the first 17 terms of this geometric series. The formula for the sum of the first \(n\) terms of a geometric series \(S_n\) is:
[tex]\[ S_n = a_1 \cdot \frac{1 - r^n}{1 - r} \][/tex]
Given:
- \(a_1 = -256\)
- \(r = -\frac{1}{4}\)
- \(n = 17\)
Plugging these values into the sum formula:
[tex]\[ S_{17} = -256 \cdot \frac{1 - \left(-\frac{1}{4}\right)^{17}}{1 - \left(-\frac{1}{4}\right)} \][/tex]
Now, let's break down the calculation:
1. Calculate \( \left(-\frac{1}{4}\right)^{17} \):
Since \(\left(-\frac{1}{4}\right)\) raised to an odd power remains negative, you would calculate:
[tex]\[ r^{17} = \left(-\frac{1}{4}\right)^{17} \][/tex]
2. Substitute \(\left(-\frac{1}{4}\right)^{17}\) in the formula:
[tex]\[ S_{17} = -256 \cdot \frac{1 - \left(-\frac{1}{4}\right)^{17}}{1 + \frac{1}{4}} \][/tex]
3. Simplify the denominator:
[tex]\[ 1 + \frac{1}{4} = \frac{5}{4} \][/tex]
4. The numerator would be:
[tex]\[ 1 - \left(-\frac{1}{4}\right)^{17} \][/tex]
Putting it all together:
[tex]\[ S_{17} = -256 \cdot \frac{1 - \left(-\frac{1}{4}\right)^{17}}{\frac{5}{4}} \][/tex]
Multiplying:
[tex]\[ S_{17} = -256 \cdot \frac{4}{5} \left(1 - \left(-\frac{1}{4}\right)^{17}\right) \][/tex]
The numerical calculation results in:
[tex]\[ S_{17} \approx -204.80000001192093 \][/tex]
Approximating this sum to the nearest tenth:
[tex]\[ S_{17} \approx -204.8 \][/tex]
Thus, the approximate sum of the first 17 terms to the nearest tenth is:
[tex]\[ \boxed{-204.8} \][/tex]
[tex]\[ a_n = a_1 \cdot r^{n-1} \][/tex]
For the given sequence:
[tex]\[ a_n = -256 \left(-\frac{1}{4}\right)^{n-1} \][/tex]
Here, \(a_1 = -256\) and \(r = -\frac{1}{4}\).
Next, we need to find the sum of the first 17 terms of this geometric series. The formula for the sum of the first \(n\) terms of a geometric series \(S_n\) is:
[tex]\[ S_n = a_1 \cdot \frac{1 - r^n}{1 - r} \][/tex]
Given:
- \(a_1 = -256\)
- \(r = -\frac{1}{4}\)
- \(n = 17\)
Plugging these values into the sum formula:
[tex]\[ S_{17} = -256 \cdot \frac{1 - \left(-\frac{1}{4}\right)^{17}}{1 - \left(-\frac{1}{4}\right)} \][/tex]
Now, let's break down the calculation:
1. Calculate \( \left(-\frac{1}{4}\right)^{17} \):
Since \(\left(-\frac{1}{4}\right)\) raised to an odd power remains negative, you would calculate:
[tex]\[ r^{17} = \left(-\frac{1}{4}\right)^{17} \][/tex]
2. Substitute \(\left(-\frac{1}{4}\right)^{17}\) in the formula:
[tex]\[ S_{17} = -256 \cdot \frac{1 - \left(-\frac{1}{4}\right)^{17}}{1 + \frac{1}{4}} \][/tex]
3. Simplify the denominator:
[tex]\[ 1 + \frac{1}{4} = \frac{5}{4} \][/tex]
4. The numerator would be:
[tex]\[ 1 - \left(-\frac{1}{4}\right)^{17} \][/tex]
Putting it all together:
[tex]\[ S_{17} = -256 \cdot \frac{1 - \left(-\frac{1}{4}\right)^{17}}{\frac{5}{4}} \][/tex]
Multiplying:
[tex]\[ S_{17} = -256 \cdot \frac{4}{5} \left(1 - \left(-\frac{1}{4}\right)^{17}\right) \][/tex]
The numerical calculation results in:
[tex]\[ S_{17} \approx -204.80000001192093 \][/tex]
Approximating this sum to the nearest tenth:
[tex]\[ S_{17} \approx -204.8 \][/tex]
Thus, the approximate sum of the first 17 terms to the nearest tenth is:
[tex]\[ \boxed{-204.8} \][/tex]
