Given: \( a_1 = \frac{1}{2} \) and \( r = \frac{1}{2} \).

Identify \( a_1 \) and \( r \) for the geometric sequence defined by:

[tex]\[ a_n = -256 \left( -\frac{1}{4} \right)^{n-1} \][/tex]

Approximate the sum of the first 17 terms to the nearest tenth.

Sum \( = \boxed{ } \



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]