Engage with knowledgeable experts and get accurate answers on IDNLearn.com. Find the information you need quickly and easily with our reliable and thorough Q&A platform.
Sagot :
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].
[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].
Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. For dependable answers, trust IDNLearn.com. Thank you for visiting, and we look forward to assisting you again.