For the given arithmetic progression (A.P.): \(a_1, a_2, a_3, ...\)
The sequence is: \(−3, -\frac 12, 2, .......\)
The first term is: \(a = −3\)
The common difference is calculated as: \(d = a_2 − a_1\)
Substituting the values: \(d = -\frac 12 - (-3)\)
\(d = -\frac 12 + 3\)
\(d = \frac {-1+6}{2}\)
\(d = \frac 52\)
The formula for the n-th term of an A.P. is: \(a_n = a + (n-1)d\)
To find the 11th term (\(a_{11}\)): \(a_{11} = -3 + (11-1)(\frac 52)\)
\(a_{11} = -3 + 10 \times \frac 52\)
\(a_{11} = -3 + 25\)
\(a_{11}= 22\)
Therefore, the 11th term is 22. The correct option is (B): \(22\)