Alright, let's tackle this problem step by step. The question is asking for the median of 27 consecutive positive integers whose sum is 3^7, which is 2187. Hmm, okay. Let me break this down.

First, I remember that the median of a set of consecutive numbers is the same as the average. Since there are 27 numbers, which is an odd count, the median will be the 14th term when they're ordered from smallest to largest. That makes sense because the middle position is (27 + 1)/2 = 14th term. So, if I can find the average of these numbers, that should give me the median.

The sum of the numbers is given as 2187. The formula for the sum of an arithmetic sequence is (number of terms) × (first term + last term)/2. But since these numbers are consecutive, the difference between each term is 1. Therefore, the average (which is the median) is just the sum divided by the number of terms. Let me check that.

Sum = average × number of terms. So, average = sum / number of terms. That means the median is 2187 / 27. Let me compute that. 2187 divided by 27. Hmm, 27 times 80 is 2160. Then 2187 - 2160 = 27. So, 27 divided by 27 is 1. Therefore, 80 + 1 = 81. So, the median should be 81.

Wait, that seems straightforward, but let me verify. If there are 27 consecutive numbers starting from n, then the numbers are n, n+1, n+2, ..., n+26. The sum of these numbers is sum = (n + (n + 26)) × 27 / 2 = (2n + 26) × 27 / 2. This should equal 2187. Let me solve for n.

So, (2n + 26) × 27 / 2 = 2187. Multiply both sides by 2: (2n + 26) × 27 = 4374. Divide both sides by 27: 2n + 26 = 4374 / 27. Let's compute 4374 divided by 27. 27 × 160 = 4320. 4374 - 4320 = 54. 54 divided by 27 is 2. So, 160 + 2 = 162. Therefore, 2n + 26 = 162. Subtract 26: 2n = 136. Divide by 2: n = 68. So, the first term is 68, and the last term is 68 + 26 = 94. The median is the 14th term. Since the numbers are consecutive, the 14th term is 68 + 13 = 81. Yep, that matches the earlier result. So, the median is indeed 81.

Alternatively, since the sum is 2187 and there are 27 numbers, the average is 2187 / 27 = 81. Since the numbers are consecutive, the average is the median. So, that's another way to see it. Therefore, the answer is 81.

But wait, let me make sure there isn't a trick here. The problem says "27 consecutive positive integers". So, starting from 68, which is positive, and ending at 94. All numbers are positive, so that's okay. The sum checks out because 27 numbers average 81 would sum to 2187. So, that's correct.

I think that's solid. The key insight is recognizing that the median of consecutive numbers is the average, which is the sum divided by the count. Since consecutive numbers form an arithmetic sequence with a common difference of 1, the average is straightforward to compute. Therefore, the median is 81.
Therefore, the median of the 27 consecutive positive integers is $\boxed{81}$.
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