To find the sum of the first 8 terms of the given geometric sequence [tex]\(1024, -256, 64, -16, 4, \ldots\)[/tex], we can use the formula for the sum of the first [tex]\(n\)[/tex] terms of a geometric sequence:
[tex]\[
S_n = \frac{a(1 - r^n)}{1 - r}
\][/tex]
where:
- [tex]\(a\)[/tex] is the first term of the sequence,
- [tex]\(r\)[/tex] is the common ratio,
- [tex]\(n\)[/tex] is the number of terms.
1. Identify the first term ([tex]\(a\)[/tex]):
The first term of the sequence is [tex]\(a = 1024\)[/tex].
2. Determine the common ratio ([tex]\(r\)[/tex]):
To find the common ratio, divide the second term by the first term:
[tex]\[
r = \frac{-256}{1024} = -0.25
\][/tex]
3. Determine the number of terms ([tex]\(n\)[/tex]):
We are given that we need to find the sum of the first 8 terms, so [tex]\(n = 8\)[/tex].
4. Substitute the known values into the formula:
[tex]\[
S_8 = \frac{1024(1 - (-0.25)^8)}{1 - (-0.25)}
\][/tex]
5. Calculate [tex]\((-0.25)^8\)[/tex]:
[tex]\[
(-0.25)^8 = 0.0000152587890625
\][/tex]
6. Substitute this value back into the formula:
[tex]\[
S_8 = \frac{1024 \left(1 - 0.0000152587890625\right)}{1 + 0.25}
\][/tex]
7. Simplify the numerator and denominator:
[tex]\[
S_8 = \frac{1024 \cdot 0.9999847412109375}{1.25}
\][/tex]
8. Perform the multiplication:
[tex]\[
1024 \cdot 0.9999847412109375 \approx 1023.984375
\][/tex]
9. Divide by 1.25:
[tex]\[
\frac{1023.984375}{1.25} = 819.1875
\][/tex]
10. Round the result to the nearest hundredth:
[tex]\[
819.19
\][/tex]
So, the sum of the first 8 terms of the sequence rounded to the nearest hundredth is [tex]\(819.19\)[/tex].
