Final answer:
The sum of the first eight terms in the series 2, 10, 50, 250, ... is -195,312 (Option d).
Step-by-step explanation:
To find the sum of the first eight terms in the series 2, 10, 50, 250, ... , we need to determine the pattern. We can observe that each term is obtained by multiplying the previous term by 5. So, the series can be written as 2, 2 * 5, (2 * 5) * 5, ((2 * 5) * 5) * 5, ... .
Using this pattern, we can find the eighth term by multiplying the first term (2) by 5 raised to the power of (8-1). This gives us the value of the eighth term as 2 * (5^7).
Now, to find the sum of the first eight terms, we can use the formula for the sum of a geometric series: S = a(1 - r^n) / (1 - r), where S is the sum, a is the first term, r is the common ratio, and n is the number of terms. Plugging in the values, we get:
S = 2 * (1 - 5^8) / (1 - 5) = 2 * (1 - 390625) / -4 = (1 - 390625) / -2 = -195312
Therefore, the sum of the first eight terms in the series is -195,312.