If the average of the first n numbers in the sequence 148, 146, 144,….., is 125, then n =

Options

A. 18
B. 24
C. 30
D. 36

Answer

To solve the problem step by step, we need to find the value of \( n \) such that the average of the first \( n \) numbers in the sequence \( 148, 146, 144, \ldots \) is \( 125 \).

### Step 1: Identify the sequence type and parameters
The given sequence is \( 148, 146, 144, \ldots \). This is an arithmetic progression (AP) where:
- The first term \( a = 148 \)
- The common difference \( d = 146 - 148 = -2 \)

**Hint:** Recognize that the sequence is an arithmetic progression and identify the first term and common difference.

### Step 2: Use the formula for the sum of the first \( n \) terms of an AP
The sum \( S_n \) of the first \( n \) terms of an arithmetic progression can be calculated using the formula:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
Substituting the values of \( a \) and \( d \):
\[ S_n = \frac{n}{2} \times (2 \times 148 + (n - 1)(-2)) \]

**Hint:** Remember the formula for the sum of the first \( n \) terms in an arithmetic progression.

### Step 3: Simplify the sum formula
Now, simplify the expression:
\[ S_n = \frac{n}{2} \times (296 - 2(n - 1)) \]
\[ = \frac{n}{2} \times (296 - 2n + 2) \]
\[ = \frac{n}{2} \times (298 - 2n) \]
\[ = n \times (149 - n) \]

**Hint:** Combine like terms and factor out \( n \) to simplify the sum.

### Step 4: Set up the average equation
The average \( A \) of the first \( n \) terms is given by:
\[ A = \frac{S_n}{n} \]
We know from the problem that the average is \( 125 \):
\[ 125 = \frac{S_n}{n} = 149 - n \]

**Hint:** Recall that the average is the sum divided by the number of terms.

### Step 5: Solve for \( n \)
Now, we can set up the equation:
\[ 125 = 149 - n \]
Rearranging gives:
\[ n = 149 - 125 \]
\[ n = 24 \]

**Hint:** Isolate \( n \) in the equation to find its value.

### Conclusion
Thus, the value of \( n \) is \( 24 \).

**Final Answer:** \( n = 24 \)