To determine the sum of the first eight terms in the geometric series [tex]\(10, 50, 250, \ldots\)[/tex], we follow this step-by-step procedure:
1. Identify the first term ([tex]\(a_1\)[/tex]) and the common ratio ([tex]\(r\)[/tex]):
- The first term [tex]\(a_1\)[/tex] in the series is [tex]\(10\)[/tex].
- The common ratio [tex]\(r\)[/tex] is found by dividing the second term by the first term: [tex]\(r = \frac{50}{10} = 5\)[/tex].
2. Determine the number of terms ([tex]\(n\)[/tex]):
- The number of terms [tex]\(n\)[/tex] we need the sum for is [tex]\(8\)[/tex].
3. Use the formula for the sum of the first [tex]\(n\)[/tex] terms of a geometric series:
For a geometric series, the sum of the first [tex]\(n\)[/tex] terms ([tex]\(S_n\)[/tex]) is given by:
[tex]\[
S_n = \frac{a_1(1 - r^n)}{1 - r}
\][/tex]
4. Substitute the known values into the formula:
- [tex]\(a_1 = 10\)[/tex]
- [tex]\(r = 5\)[/tex]
- [tex]\(n = 8\)[/tex]
Thus, the sum [tex]\(S_8\)[/tex] is:
[tex]\[
S_8 = \frac{10(1 - 5^8)}{1 - 5}
\][/tex]
5. Calculate the denominator and numerator separately:
- The denominator [tex]\(1 - 5 = -4\)[/tex].
- The numerator [tex]\(10(1 - 5^8)\)[/tex].
6. Resolve [tex]\(5^8\)[/tex]:
- [tex]\(5^8 = 390,625\)[/tex]
7. Continue the calculation:
- The numerator becomes [tex]\(10(1 - 390,625) = 10(-390,624) = -3,906,240\)[/tex].
8. Finally, divide the numerator by the denominator:
[tex]\[
S_8 = \frac{-3,906,240}{-4} = 976,560
\][/tex]
Therefore, the sum of the first eight terms in the series [tex]\(10, 50, 250, \ldots\)[/tex] is [tex]\(976,560\)[/tex].
So, the correct answer is not listed in the provided options. However, based on our calculations, [tex]\(976,560\)[/tex] is the accurate sum.
