The 15th term of the series `2 1/2+1 7/(13)+1 1/9+(20)/(23)+..` is

Options

A. `10/39`
B. `10/21`
C. `10/23`
D. none of these

Explanation

To find the 15th term of the series \(2 \frac{1}{2} + 1 \frac{7}{13} + 1 \frac{1}{9} + \frac{20}{23} + \ldots\), we will analyze the series step by step.

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we convert the mixed numbers in the series to improper fractions:
- \(2 \frac{1}{2} = \frac{5}{2}\)
- \(1 \frac{7}{13} = \frac{20}{13}\)
- \(1 \frac{1}{9} = \frac{10}{9}\)

So, the series can be rewritten as:
\[ \frac{5}{2}, \frac{20}{13}, \frac{10}{9}, \frac{20}{23}, \ldots \]

### Step 2: Identify the Pattern
Next, we observe the numerators and denominators:
- The numerators are \(5, 20, 10, 20, \ldots\)
- The denominators are \(2, 13, 9, 23, \ldots\)

### Step 3: Analyze the Denominators
The denominators appear to follow a pattern. Let's analyze them:
- The denominators are \(2, 13, 9, 23\).
- We can see that the denominators can be expressed as:
- \(2 = 2 + 0 \cdot 5\)
- \(13 = 8 + 1 \cdot 5\)
- \(9 = 4 + 1 \cdot 5\)
- \(23 = 18 + 1 \cdot 5\)

We can see that the denominators are in an arithmetic progression (AP) with a common difference of \(5\).

### Step 4: General Form of the Denominators
The general term for the denominators can be expressed as:
\[ D_n = 2 + (n-1) \cdot 5 \]
This simplifies to:
\[ D_n = 5n - 3 \]

### Step 5: General Form of the Numerators
The numerators seem to alternate between \(5\) and \(20\). We can express the numerators as:
- For odd \(n\), the numerator is \(20\).
- For even \(n\), the numerator is \(5\).

Thus, we can define the general term for the numerators as:
\[ N_n = \begin{cases} 20 & \text{if } n \text{ is odd} \\ 5 & \text{if } n \text{ is even} \end{cases} \]

### Step 6: Finding the 15th Term
Now, we can find the 15th term of the series:
- Since \(15\) is odd, we use the numerator for odd \(n\):
\[ N_{15} = 20 \]
- The denominator for \(n = 15\):
\[ D_{15} = 5(15) - 3 = 75 - 3 = 72 \]

### Step 7: Combine Numerator and Denominator
Thus, the 15th term of the series is:
\[ T_{15} = \frac{N_{15}}{D_{15}} = \frac{20}{72} \]

### Step 8: Simplify the Fraction
We can simplify \(\frac{20}{72}\):
\[ \frac{20}{72} = \frac{5}{18} \]

### Final Answer
The 15th term of the series is:
\[ \frac{5}{18} \]