Okay, let's see. I need to find the number of terms, n, in the geometric sequence {1/4, 1/8, 1/16, ...} such that the sum of the first n terms is 255/512. Hmm, geometric sequences have a common ratio between consecutive terms. Let me recall the formula for the sum of the first n terms of a geometric series.

First, identify the first term and the common ratio. The first term a is 1/4. The common ratio r is the factor by which each term is multiplied to get the next term. Let me check: 1/8 divided by 1/4 is (1/8)*(4/1) = 1/2. Similarly, 1/16 divided by 1/8 is 1/2. So, yes, the common ratio r is 1/2.

The formula for the sum S_n of the first n terms of a geometric series is S_n = a*(1 - r^n)/(1 - r), right? Since r ≠ 1. Let me confirm that. If r is not 1, the formula holds. If r were 1, the sum would just be a*n, but here r is 1/2, so we're good.

Given that S_n = 255/512, and we need to solve for n. Let's plug in the values we have into the formula:

255/512 = (1/4)*(1 - (1/2)^n)/(1 - 1/2)

Simplify the denominator first: 1 - 1/2 = 1/2. So the formula becomes:

255/512 = (1/4)*(1 - (1/2)^n)/(1/2)

Dividing by 1/2 is the same as multiplying by 2, so:

255/512 = (1/4)*2*(1 - (1/2)^n)

Simplify 1/4 * 2 = 1/2. So:

255/512 = (1/2)*(1 - (1/2)^n)

Multiply both sides by 2 to eliminate the 1/2:

255/256 = 1 - (1/2)^n

Now, subtract 255/256 from both sides:

(1/2)^n = 1 - 255/256

Calculate 1 - 255/256. 1 is 256/256, so 256/256 - 255/256 = 1/256. Therefore:

(1/2)^n = 1/256

Now, 1/256 is equal to (1/2)^8, because 2^8 = 256. Let me check: 2^1=2, 2^2=4, 2^3=8, 2^4=16, 2^5=32, 2^6=64, 2^7=128, 2^8=256. Yes, so 1/256 is (1/2)^8. Therefore:

n = 8

Wait, but let me verify this. Let's calculate the sum of the first 8 terms of the series.

First term a = 1/4, common ratio r = 1/2, n = 8.

Sum S_8 = (1/4)*(1 - (1/2)^8)/(1 - 1/2) = (1/4)*(1 - 1/256)/(1/2) = (1/4)*(255/256)/(1/2) = (1/4)*(255/256)*2 = (1/4)*(255/128) = 255/512. Yes, that's correct. So n is indeed 8.

But wait, let me check if maybe there was a miscalculation earlier. Let's go through the steps again.

Starting with the sum formula:

S_n = a*(1 - r^n)/(1 - r)

Given S_n = 255/512, a = 1/4, r = 1/2.

Plug in the values:

255/512 = (1/4)*(1 - (1/2)^n)/(1 - 1/2)

Simplify denominator: 1 - 1/2 = 1/2. So:

255/512 = (1/4)*(1 - (1/2)^n)/(1/2) = (1/4)*(2)*(1 - (1/2)^n) = (1/2)*(1 - (1/2)^n)

Then multiply both sides by 2:

255/256 = 1 - (1/2)^n

Therefore, (1/2)^n = 1 - 255/256 = 1/256

Which gives n = 8, since (1/2)^8 = 1/256.

Yes, that seems correct. So n is 8. Wait, but the problem statement says "the first n terms in the infinite geometric sequence {1/4, 1/8, 1/16, ...}". But this is an infinite sequence, but we are taking the sum of the first n terms. So even though the sequence is infinite, we are only summing up to the nth term, which is finite. So the formula applies here.

Alternatively, maybe the problem is expecting a different approach? Let me check the terms:

First term: 1/4 = 0.25

Second term: 1/8 = 0.125

Third term: 1/16 = 0.0625

Fourth term: 1/32 = 0.03125

Fifth term: 1/64 = 0.015625

Sixth term: 1/128 = 0.0078125

Seventh term: 1/256 = 0.00390625

Eighth term: 1/512 = 0.001953125

If we sum these up:

0.25 + 0.125 = 0.375

+ 0.0625 = 0.4375

+ 0.03125 = 0.46875

+ 0.015625 = 0.484375

+ 0.0078125 = 0.4921875

+ 0.00390625 = 0.49609375

+ 0.001953125 = 0.498046875

Wait, 0.498046875 is equal to 255/512. Let me check: 255 divided by 512.

255 ÷ 512. Let's compute:

512 × 0.5 = 256, so 255 is 1 less than 256, so 255/512 = (256 - 1)/512 = 0.5 - 1/512 ≈ 0.498046875. Yes, exactly. So the sum after 8 terms is indeed 255/512. Therefore, n is 8.

Therefore, the answer is 8. But let me check if there's a different interpretation of the problem. Wait, the problem says "the sum of the first n terms in the infinite geometric sequence". So maybe someone might think that the sum of the infinite series is being considered, but in this case, the sum of the infinite series would be a/(1 - r) = (1/4)/(1 - 1/2) = (1/4)/(1/2) = 1/2. But the problem states the sum is 255/512 ≈ 0.498, which is just short of 1/2. So the sum of the first n terms approaches 1/2 as n increases. So indeed, 255/512 is just 1/2 minus 1/512, which is 255/512. So that's the sum after 8 terms, as we found.

Therefore, n is 8. So the answer is 8.

But wait, let me check once more. Let me compute the sum step by step manually:

First term: 1/4 = 0.25

Sum after 1 term: 0.25

Sum after 2 terms: 0.25 + 0.125 = 0.375

Sum after 3 terms: 0.375 + 0.0625 = 0.4375

Sum after 4 terms: 0.4375 + 0.03125 = 0.46875

Sum after 5 terms: 0.46875 + 0.015625 = 0.484375

Sum after 6 terms: 0.484375 + 0.0078125 = 0.4921875

Sum after 7 terms: 0.4921875 + 0.00390625 = 0.49609375

Sum after 8 terms: 0.49609375 + 0.001953125 = 0.498046875, which is 255/512. So yes, that's correct. Therefore, n is 8.

Alternatively, maybe there's a formulaic way to check. Let's use the formula again:

S_n = (1/4)*(1 - (1/2)^n)/(1 - 1/2) = (1/4)*(1 - (1/2)^n)/(1/2) = (1/4)*(2)*(1 - (1/2)^n) = (1/2)*(1 - (1/2)^n)

Set this equal to 255/512:

(1/2)*(1 - (1/2)^n) = 255/512

Multiply both sides by 2:

1 - (1/2)^n = 255/256

Therefore, (1/2)^n = 1 - 255/256 = 1/256

Which gives n = log_{1/2}(1/256) = log_{2}(256) = 8, since (1/2)^8 = 1/256. Therefore, n = 8.

Yes, so all methods lead to n = 8. Therefore, the answer is 8.
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