<Question>
The sum of the first $n$ terms in the infinite geometric sequence $\left\{\frac{1}{4},\frac{1}{8},\frac{1}{16},\dots \right\}$ is $\frac{255}{512}$. Find $n$.
</Question>

<think>
<Plan_1>
Okay, so 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. Let me start by recalling the formula for the sum of the first n terms of a geometric series.

<prompt_1.1> Recall the formula for the sum of the first n terms of a geometric series and identify the first term and common ratio from the given sequence. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
The formula is S_n = a * (1 - r^n) / (1 - r), where a is the first term, r is the common ratio, and n is the number of terms. Let me confirm that. Yeah, that seems right. This formula applies when the common ratio is not equal to 1. In this case, since the sequence is geometric, each term is multiplied by a constant ratio.

Looking at the given sequence: the first term a is 1/4. The next term is 1/8, then 1/16, and so on. To find the common ratio r, I can divide the second term by the first term: (1/8) / (1/4) = (1/8) * (4/1) = 4/8 = 1/2. Let me check the next term: (1/16) / (1/8) = (1/16) * (8/1) = 8/16 = 1/2. So the common ratio r is 1/2.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Now, the sum of the first n terms is given as 255/512. So plugging into the formula: S_n = (1/4) * (1 - (1/2)^n) / (1 - 1/2) = 255/512. Let me simplify the denominator first.
The denominator is 1 - 1/2 = 1/2. So the formula becomes (1/4) * (1 - (1/2)^n) / (1/2). Dividing by 1/2 is the same as multiplying by 2, so this simplifies to (1/4) * 2 * (1 - (1/2)^n).

<prompt_2.1> Simplify the expression for S_n using the formula and set it equal to 255/512. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
Calculating that, 1/4 * 2 = 1/2. So S_n = (1/2) * (1 - (1/2)^n). And this is equal to 255/512. So now, set up the equation: (1/2)(1 - (1/2)^n) = 255/512.
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Let me solve for n. First, multiply both sides by 2 to eliminate the 1/2.
That gives: 1 - (1/2)^n = 510/512.

<prompt_3.1> Solve the equation for n by simplifying and taking logarithms. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
Wait, 255/512 * 2 = 510/512. Simplify 510/512: both numerator and denominator are divisible by 2. 510 ÷ 2 = 255, 512 ÷ 2 = 256. So 255/256. Therefore, 1 - (1/2)^n = 255/256.

Then, subtract 255/256 from both sides: 1 - 255/256 = (1/2)^n. Calculating the left side: 1 is 256/256, so 256/256 - 255/256 = 1/256. Therefore, (1/2)^n = 1/256.

Now, I need to solve for n in the equation (1/2)^n = 1/256. Since 1/256 is a power of 1/2, I can express 1/256 as (1/2)^n. Let's see, 2^8 = 256, so (1/2)^8 = 1/256. Therefore, n must be 8.
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
Let me check if that's correct. Let's verify by calculating the sum of the first 8 terms.

<prompt_4.1> Verify the solution by calculating the sum of the first 8 terms and checking if it equals 255/512. </prompt_4.1>

But wait, let me just go through the steps again to make sure I didn't make any mistakes. Starting from the beginning: the first term is 1/4, ratio is 1/2.

<prompt_4.2> Re-evaluate the solution by going through the steps again to ensure no mistakes were made. </prompt_4.2>

Alternatively, maybe there's another way to approach this problem. Let me think. The sum S_n = a1*(r^n -1)/(r -1), but since r is 1/2, which is less than 1, the formula is adjusted.

<prompt_4.3> Explore an alternative approach to solve the problem using a different formula for the sum of a geometric series. </prompt_4.3>

Alternatively, maybe I can think in terms of binary fractions. Since each term is a power of 1/2, but scaled by 1/4. Let's see, 1/4 is 2^-2, 1/8 is 2^-3, 1/16 is 2^-4, etc. So the sum S_n = 2^-2 + 2^-3 + 2^-4 + ... + 2^{-(n+1)}. This is a geometric series with first term 2^-2 and ratio 1/2. The sum would be (2^-2)*(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), which is the same as before. So same result.
</Plan_4>

<Execution_4>
<execution_4.1>
The terms would be 1/4, 1/8, 1/16, 1/32, 1/64, 1/128, 1/256, 1/512. Let's add them up step by step. 

First term: 1/4 = 0.25

Second term: 0.25 + 0.125 = 0.375

Third term: 0.375 + 0.0625 = 0.4375

Fourth term: 0.4375 + 0.03125 = 0.46875

Fifth term: 0.46875 + 0.015625 = 0.484375

Sixth term: 0.484375 + 0.0078125 = 0.4921875

Seventh term: 0.4921875 + 0.00390625 = 0.49609375

Eighth term: 0.49609375 + 0.001953125 = 0.498046875

Now, converting 255/512 to decimal: 255 ÷ 512. Let's compute that. 512 goes into 255 zero times. 512 goes into 2550 four times (4*512=2048), remainder 502. Bring down next 0: 5020. 512*9=4608, remainder 412. Bring down 0: 4120. 512*8=4096, remainder 24. Bring down 0: 240. 512 goes into 240 zero times. So approximately 0.498046875. Which matches the sum after 8 terms. So that checks out.
</execution_4.1>

<execution_4.2>
The sum formula is S_n = a*(1 - r^n)/(1 - r). Plugging in a = 1/4, r = 1/2. So S_n = (1/4)*(1 - (1/2)^n)/(1 - 1/2). Denominator is 1/2, so dividing by 1/2 is multiplying by 2. So (1/4)*2*(1 - (1/2)^n) = (1/2)*(1 - (1/2)^n). Set equal to 255/512. Then multiply both sides by 2: 1 - (1/2)^n = 510/512. Wait, 510/512 simplifies to 255/256. So 1 - (1/2)^n = 255/256. Then (1/2)^n = 1 - 255/256 = 1/256. So (1/2)^n = 1/256. Since 256 is 2^8, so (1/2)^n = (1/2)^8, so n=8.
</execution_4.2>

<execution_4.3>
Wait, actually, the formula I used is correct. For |r| < 1, S_n = a*(1 - r^n)/(1 - r). So substituting a = 1/4, r = 1/2. Then 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). Then setting this equal to 255/512. So, 1/2*(1 - (1/2)^n) = 255/512. Multiply both sides by 2: 1 - (1/2)^n = 510/512 = 255/256. Then (1/2)^n = 1 - 255/256 = 1/256. Since 1/256 is (1/2)^8, so n=8.
</execution_4.3>
</Execution_4>

<Plan_5>
Based on execution_4.3:
I don't see any mistakes in the calculations. The key steps were identifying the first term and common ratio correctly, applying the sum formula, solving the resulting equation, and verifying the solution.
</Plan_5>

<Execution_5>
No prompt to execute.
</Execution_5>

<Final_answer>
Therefore, the value of $n$ is $\boxed{8}$.
</Final_answer>
</think>