Final answer:
To determine how many terms of the geometric series 1, 2, 4, 8, ..., are needed to exceed a sum of 500, the smallest power of 2 greater than 501 is found, which is 2^9 = 512, indicating that 9 terms are required.
Step-by-step explanation:
The question asks how many terms of the series 1, 2, 4, 8, ..., must be added together for the sum to exceed 500. This is a geometric series where each term is double the preceding term, also known as a sequence in which the ratio between successive terms is constant.
To find how many terms are needed to exceed 500, we can use the formula for the sum of the first n terms of a geometric series, which is given by Sn = a1(1 - rn)/(1 - r), where a1 is the first term and r is the common ratio. Here, a1 is 1 and r is 2. We want to find the smallest n such that Sn > 500.
We set up the inequality 1(1 - 2n)/(1 - 2) > 500 and solve for n. Simplifying the inequality, we get 2n - 1 > 500 . Therefore, n must satisfy 2n > 501.
Testing powers of 2, we find that 29 = 512, which is the smallest power of 2 greater than 501.
Thus, 9 terms are needed for the sum to exceed 500.