Find the 50th term of the sequence
`1/n, (n+1)/(n), (2n+1)/(n),,……..`

Answer

To find the 50th term of the sequence given by \( \frac{1}{n}, \frac{n+1}{n}, \frac{2n+1}{n}, \ldots \), we will follow these steps:

### Step 1: Identify the first term and the common difference
The first term of the sequence, denoted as \( a \), is:
\[ a = \frac{1}{n} \]

Next, we need to find the common difference \( d \). The second term is \( \frac{n+1}{n} \), so we calculate the common difference as follows:
\[ d = \text{second term} - \text{first term} = \frac{n+1}{n} - \frac{1}{n} \]
We can simplify this:
\[ d = \frac{n + 1 - 1}{n} = \frac{n}{n} = 1 \]

### Step 2: Use the formula for the nth term of an arithmetic progression
The formula for the \( n \)-th term of an arithmetic progression is given by:
\[ a_n = a + (n - 1) \cdot d \]
To find the 50th term, we substitute \( n = 50 \):
\[ a_{50} = a + (50 - 1) \cdot d \]

### Step 3: Substitute the values of \( a \) and \( d \)
Now we substitute \( a = \frac{1}{n} \) and \( d = 1 \) into the formula:
\[ a_{50} = \frac{1}{n} + (50 - 1) \cdot 1 \]
This simplifies to:
\[ a_{50} = \frac{1}{n} + 49 \]

### Step 4: Combine the terms
To express \( a_{50} \) in a single fraction, we take the least common multiple (LCM) of \( n \):
\[ a_{50} = \frac{1}{n} + 49 = \frac{1 + 49n}{n} \]

### Final Result
Thus, the 50th term of the sequence is:
\[ \boxed{\frac{49n + 1}{n}} \]

---