The average of 26, 29, n,35 and 43 lies between 25 and 35. If n is always an integer and greater than the average of the given integers then the value of n is :

Options

A. 33 `lt n lt` 47
B. 34 `lt n lt` 43
C. 33 `lt n lt` 42
D. none of these

Answer

To solve the problem, we need to find the integer value of \( n \) such that the average of the numbers 26, 29, \( n \), 35, and 43 lies between 25 and 35, and \( n \) is greater than this average.

### Step 1: Calculate the average of the numbers
The average of the numbers can be calculated using the formula:
\[ \text{Average} = \frac{\text{Sum of all numbers}}{\text{Total number of values}} \]
Here, the numbers are 26, 29, \( n \), 35, and 43. Therefore, the sum of these numbers is:
\[ 26 + 29 + n + 35 + 43 = 133 + n \]
The total number of values is 5. Thus, the average is:
\[ \text{Average} = \frac{133 + n}{5} \]

### Step 2: Set up the inequality for the average
We know that the average lies between 25 and 35. Therefore, we can set up the following inequalities:
\[ 25 < \frac{133 + n}{5} < 35 \]

### Step 3: Solve the inequalities
First, we will solve the left part of the inequality:
\[ 25 < \frac{133 + n}{5} \]
Multiplying both sides by 5 gives:
\[ 125 < 133 + n \]
Subtracting 133 from both sides results in:
\[ n > -8 \]
This inequality is not restrictive since \( n \) is a positive integer.

Now, we will solve the right part of the inequality:
\[ \frac{133 + n}{5} < 35 \]
Multiplying both sides by 5 gives:
\[ 133 + n < 175 \]
Subtracting 133 from both sides results in:
\[ n < 42 \]

### Step 4: Combine the results
From the inequalities we have:
\[ -8 < n < 42 \]
Since \( n \) must be an integer, we can simplify this to:
\[ n \in \{0, 1, 2, \ldots, 41\} \]

### Step 5: Consider the condition that \( n \) is greater than the average
We also know that \( n \) must be greater than the average. The average is:
\[ \frac{133 + n}{5} \]
So we set up the inequality:
\[ n > \frac{133 + n}{5} \]
Multiplying both sides by 5 gives:
\[ 5n > 133 + n \]
Subtracting \( n \) from both sides results in:
\[ 4n > 133 \]
Dividing both sides by 4 gives:
\[ n > 33.25 \]
Since \( n \) must be an integer, we can conclude:
\[ n \geq 34 \]

### Step 6: Final range for \( n \)
Combining the results from the previous steps, we have:
\[ 34 \leq n < 42 \]
The possible integer values for \( n \) are:
\[ n \in \{34, 35, 36, 37, 38, 39, 40, 41\} \]

### Step 7: Conclusion
The integer values of \( n \) that satisfy all conditions are 34, 35, 36, 37, 38, 39, 40, and 41. However, since \( n \) must be greater than the average, we check the maximum value of \( n \) that keeps the average below 35.

The only integer value that fits all conditions is:
\[ \boxed{34} \]