Choose the correct choice in the following and justify :
`11^(th)` term of the `A . P .: -3- (1)/(2), 2 ,….` is

Options

A. 28
B. 22
C. `-38`
D. `- 48 (1)/(2)`

Answer

To find the 11th term of the arithmetic progression (A.P.) given by the sequence -3, -1/2, 2, ..., we will follow these steps:

### Step 1: Identify the first term (a) and the common difference (d)
The first term \( a \) of the A.P. is:
\[ a = -3 \]

Next, we need to find the common difference \( d \). The common difference is calculated by subtracting the first term from the second term:
\[ d = \text{second term} - \text{first term} = -\frac{1}{2} - (-3) \]
Calculating this gives:
\[ d = -\frac{1}{2} + 3 = -\frac{1}{2} + \frac{6}{2} = \frac{5}{2} \]

### Step 2: Use the formula for the nth term of an A.P.
The formula for the nth term of an A.P. is given by:
\[ a_n = a + (n-1)d \]
We need to find the 11th term, so we set \( n = 11 \):
\[ a_{11} = a + (11-1)d = a + 10d \]

### Step 3: Substitute the values of \( a \) and \( d \)
Substituting the values we found:
\[ a_{11} = -3 + 10 \times \frac{5}{2} \]
Calculating \( 10 \times \frac{5}{2} \):
\[ 10 \times \frac{5}{2} = 25 \]
Now substituting this back into the equation:
\[ a_{11} = -3 + 25 \]

### Step 4: Perform the final calculation
Calculating the final result:
\[ a_{11} = 22 \]

### Conclusion
Thus, the 11th term of the A.P. is:
\[ \boxed{22} \]