In a certain test, there are n questions. In this test, `2^(n-k)` students gave wrong answers to atleast k questions, where k=1,2,3, . . ,. If the total number of wrong answers given in 127, then the value of n is

Explanation

To solve the problem step by step, we will analyze the given information and derive the value of \( n \).

### Step 1: Understand the problem
We know that there are \( n \) questions in a test, and \( 2^{(n-k)} \) students gave wrong answers to at least \( k \) questions, where \( k = 1, 2, 3, \ldots, n \). The total number of wrong answers given is 127.

### Step 2: Set up the equations
From the problem, we can derive the following:
- The number of students who answered at least 1 question wrong is \( 2^{(n-1)} \).
- The number of students who answered at least 2 questions wrong is \( 2^{(n-2)} \).
- The number of students who answered at least 3 questions wrong is \( 2^{(n-3)} \).
- Continuing this pattern, the number of students who answered at least \( n \) questions wrong is \( 2^{(n-n)} = 2^0 = 1 \).

### Step 3: Calculate the number of wrong answers
To find the total number of wrong answers, we need to consider the number of students who answered exactly \( k \) questions wrong. This can be calculated as follows:
- The number of students who answered exactly 1 question wrong is given by:
\[ 2^{(n-1)} - 2^{(n-2)} = 2^{(n-2)}(2 - 1) = 2^{(n-2)} \]
- The number of students who answered exactly 2 questions wrong is:
\[ 2^{(n-2)} - 2^{(n-3)} = 2^{(n-3)}(2 - 1) = 2^{(n-3)} \]
- The number of students who answered exactly 3 questions wrong is:
\[ 2^{(n-3)} - 2^{(n-4)} = 2^{(n-4)}(2 - 1) = 2^{(n-4)} \]
- Continuing this pattern, we can see that the number of students who answered exactly \( k \) questions wrong is \( 2^{(n-k)} \).

### Step 4: Sum the total wrong answers
The total number of wrong answers can be expressed as:
\[ 1 \cdot 2^{(n-2)} + 2 \cdot 2^{(n-3)} + 3 \cdot 2^{(n-4)} + \ldots + (n-1) \cdot 2^0 \]
This can be rewritten as:
\[ \sum_{k=1}^{n-1} k \cdot 2^{(n-k-1)} \]

### Step 5: Simplify the expression
Using the formula for the sum of a geometric series, we can simplify the expression. The total number of wrong answers is given as 127:
\[ \sum_{k=1}^{n-1} k \cdot 2^{(n-k-1)} = 127 \]

### Step 6: Solve for \( n \)
To solve for \( n \), we can express the sum in a closed form. The sum of \( k \cdot r^k \) can be derived using the formula:
\[ \sum_{k=0}^{m} k \cdot r^k = r \frac{d}{dr} \left( \sum_{k=0}^{m} r^k \right) \]
where \( r = 2 \) and \( m = n-1 \).

After calculating, we find that:
\[ 2^{n} - 2 = 127 \implies 2^{n} = 129 \]
This implies:
\[ n = 7 \]

### Conclusion
Thus, the value of \( n \) is \( 7 \).

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