To determine the sum of the first 100 terms of the sequence where the [tex]\( n \)[/tex]th term is given by [tex]\( t_n = 200^n \)[/tex], we can recognize this as a geometric series. The formula for the sum of the first [tex]\( n \)[/tex] terms of a geometric series is:
[tex]\[ S_n = a \frac{r^n - 1}{r - 1} \][/tex]
where:
- [tex]\( a \)[/tex] is the first term of the series,
- [tex]\( r \)[/tex] is the common ratio,
- [tex]\( n \)[/tex] is the number of terms.
Given the sequence [tex]\( t_n = 200^n \)[/tex], we can identify:
- The first term [tex]\( t_1 = 200^1 = 200 \)[/tex] (so [tex]\( a = 200 \)[/tex]),
- The common ratio [tex]\( r = 200 \)[/tex],
- The number of terms [tex]\( n = 100 \)[/tex].
Utilizing the formula:
[tex]\[ S_{100} = 200 \frac{200^{100} - 1}{200 - 1} \][/tex]
Simplifying the denominator:
[tex]\[ S_{100} = 200 \frac{200^{100} - 1}{199} \][/tex]
Thus, the correct expression that represents the sum of the first 100 terms is:
[tex]\[ \boxed{\frac{200 \left( 200^{100} - 1 \right)}{199}} \][/tex]
Given the numeric result from the formula, the computed sum of the first 100 terms in this geometric sequence is [tex]\( 1.2740207037469641 \times 10^{230} \)[/tex]. This reaffirms that option A is the correct choice.
