The median of the numbers `n+3, n+7, …. N + 31, n + 35` is 30. What is the value of n ?

Options

A. 11
B. 14
C. 19
D. 27

Explanation

To find the value of \( n \) given that the median of the numbers \( n + 3, n + 7, \ldots, n + 31, n + 35 \) is 30, we can follow these steps:

### Step 1: Identify the sequence of numbers
The numbers given are:
- \( n + 3 \)
- \( n + 7 \)
- \( n + 11 \)
- \( n + 15 \)
- \( n + 19 \)
- \( n + 23 \)
- \( n + 27 \)
- \( n + 31 \)
- \( n + 35 \)

This forms an arithmetic sequence where the first term is \( n + 3 \) and the common difference is 4.

### Step 2: Count the number of terms
To find the number of terms in the sequence, we can use the formula for the \( k \)-th term of an arithmetic sequence:
\[ a_k = a + (k-1)d \]
Where:
- \( a \) is the first term
- \( d \) is the common difference
- \( k \) is the number of terms

Setting \( a_k = n + 35 \):
\[ n + 35 = n + 3 + (k-1) \cdot 4 \]
Simplifying this:
\[ n + 35 = n + 3 + 4k - 4 \]
\[ n + 35 = n - 1 + 4k \]
\[ 36 = 4k \]
\[ k = 9 \]

### Step 3: Find the median
Since there are 9 terms (an odd number), the median will be the middle term, which is the 5th term in the sequence.

The 5th term can be calculated as:
\[ \text{5th term} = n + 3 + 4 \cdot 4 = n + 3 + 16 = n + 19 \]

### Step 4: Set the median equal to 30
We know the median is given as 30:
\[ n + 19 = 30 \]

### Step 5: Solve for \( n \)
To find \( n \), we can rearrange the equation:
\[ n = 30 - 19 \]
\[ n = 11 \]

### Final Answer
The value of \( n \) is \( 11 \).

---