Find the 30th term of the sequence: `1/2 , 1, 3/2`,….

Answer

To find the 30th term of the sequence \( \frac{1}{2}, 1, \frac{3}{2}, \ldots \), we can follow these steps:

### Step 1: Identify the first term and common difference
The first term \( a \) of the sequence is:
\[ a = \frac{1}{2} \]
The second term is \( 1 \), and we can find the common difference \( d \) by subtracting the first term from the second term:
\[ d = 1 - \frac{1}{2} = \frac{1}{2} \]

### Step 2: Write the formula for the nth term of an arithmetic progression
The formula for the nth term \( a_n \) of an arithmetic progression is given by:
\[ a_n = a + (n - 1) \cdot d \]

### Step 3: Substitute values to find the 30th term
To find the 30th term \( a_{30} \), we substitute \( n = 30 \), \( a = \frac{1}{2} \), and \( d = \frac{1}{2} \) into the formula:
\[ a_{30} = a + (30 - 1) \cdot d \]
\[ a_{30} = \frac{1}{2} + (29) \cdot \frac{1}{2} \]

### Step 4: Simplify the expression
Now we simplify the expression:
\[ a_{30} = \frac{1}{2} + \frac{29}{2} \]
To add these fractions, we can combine them:
\[ a_{30} = \frac{1 + 29}{2} = \frac{30}{2} = 15 \]

### Final Answer
Thus, the 30th term of the given sequence is:
\[ \boxed{15} \]