An arithmetic series consists of 2n terms, and the first term equals the value of the common difference. If a new series is formed taking the 1st, 3rd, 5th,... (2n – 1) th term of the old series, find the ratio of the sum of the new series to that of the sum of the terms of the old series.

Options

A. `(n+1)/(2(2n+1))`
B. `(n)/(2n +1)`
C. `1/2`
D. cannot be determined

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Define the Arithmetic Series
Let the first term of the arithmetic series be \( a \) and the common difference be \( d \). According to the problem, we have:
- The number of terms in the series is \( 2n \).
- The first term equals the common difference, so \( a = d \).

### Step 2: Write the Terms of the Old Series
The terms of the arithmetic series can be expressed as:
- \( a, a + d, a + 2d, \ldots, a + (2n - 1)d \)

Since \( a = d \), we can rewrite the terms as:
- \( a, 2a, 3a, \ldots, 2na \)

### Step 3: Calculate the Sum of the Old Series
The sum \( S_{old} \) of the first \( 2n \) terms of an arithmetic series can be calculated using the formula:
\[ S_{old} = \frac{n}{2} \times (2a + (n-1)d) \]
In our case:
\[ S_{old} = \frac{2n}{2} \times (2a + (2n-1)a) = n \times (2a + (2n-1)a) = n \times (2a + 2na - a) = n \times (2na + a) = n \times a(2n + 1) \]

### Step 4: Identify the New Series
The new series is formed by taking the 1st, 3rd, 5th, ..., (2n - 1)th terms of the old series. The terms of the new series are:
- \( a, 3a, 5a, \ldots, (2n-1)a \)

### Step 5: Calculate the Sum of the New Series
The number of terms in the new series is \( n \) (since we take every second term from the old series). The sum \( S_{new} \) can be calculated as follows:
\[ S_{new} = \frac{n}{2} \times (first\ term + last\ term) = \frac{n}{2} \times (a + (2n-1)a) = \frac{n}{2} \times (2na) = n^2a \]

### Step 6: Find the Ratio of the Sums
Now we need to find the ratio of the sum of the new series to the sum of the old series:
\[ \text{Ratio} = \frac{S_{new}}{S_{old}} = \frac{n^2a}{n \cdot a(2n + 1)} = \frac{n}{2n + 1} \]

### Conclusion
Thus, the ratio of the sum of the new series to that of the sum of the terms of the old series is:
\[ \frac{n}{2n + 1} \]