The average of 1, 3, 5, 7, 9, 11, ........ to 25 terms is:

Options

A. 125
B. 25
C. 625
D. 50

Answer

To find the average of the first 25 terms of the series 1, 3, 5, 7, 9, ..., we can follow these steps:

### Step 1: Identify the series
The series given is an arithmetic progression (AP) where:
- The first term \( A = 1 \)
- The common difference \( D = 2 \)

### Step 2: Find the last term
To find the last term of the series when there are 25 terms, we can use the formula for the nth term of an AP:
\[ L = A + (N - 1) \times D \]
Where:
- \( N \) is the number of terms (25 in this case)

Substituting the values:
\[ L = 1 + (25 - 1) \times 2 \]
\[ L = 1 + 24 \times 2 \]
\[ L = 1 + 48 = 49 \]

### Step 3: Calculate the sum of the first 25 terms
The sum of the first \( N \) terms of an AP can be calculated using the formula:
\[ S_N = \frac{N}{2} \times (A + L) \]
Substituting the known values:
\[ S_{25} = \frac{25}{2} \times (1 + 49) \]
\[ S_{25} = \frac{25}{2} \times 50 \]
\[ S_{25} = 25 \times 25 = 625 \]

### Step 4: Calculate the average
The average \( \text{Average} \) is given by:
\[ \text{Average} = \frac{S_N}{N} \]
Where \( S_N \) is the sum of the terms and \( N \) is the number of terms. Thus:
\[ \text{Average} = \frac{625}{25} \]
\[ \text{Average} = 25 \]

### Final Answer
The average of the first 25 terms of the series is **25**.

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