Final answer:
The sum of the geometric sequence starting with 1600 and ending with 6.25, halving each time, is 3200. The sequence is identified by its common ratio of 1/2, allowing the use of the geometric sum formula to find the total sum.
Step-by-step explanation:
The sum of the given sequence 1600, 800, 400,…, 6.25 is 3200. This sequence is a geometric progression with a common ratio of 1/2.Explanation: To find the sum of a geometric sequence, use the formula S = a1(1 - r^n) / (1 - r), where S is the sum, a1 is the first term, r is the common ratio, and n is the number of terms. The first term a1 is 1600 and the common ratio r is 1/2. Since the last term is 6.25, we find that n = 8 (because 1600*(1/2)^(n-1) = 6.25). Plugging these values into the formula gives us the sum of the sequence.To find the sum, we need to add up all the terms in the sequence. The given sequence has a common ratio of 1/2, as each term is half of the previous term. So, we can write the sequence as
1600 + 800 + 400 + ... + 6.25Now, we can use the formula for the sum of a geometric sequence to find the sum:Sum = (first term * (1 - common ratio^(number of terms))) / (1 - common ratio)Plugging in the values gives us:Sum = (1600 * (1 - (1/2)^4)) / (1 - 1/2) = 1606.25Therefore, the sum of the given sequence is 1606.25 (option b).