Determine whether the following sequence is an $A.P.$ or not. (Assume that the pattern continues.) If it is an $A.P.$,find its $n^{th}$ term: $111, 107, 103, 99, \ldots$

(A) To determine if the sequence is an $A.P.$,we check the common difference $d = a_{n} - a_{n-1}$.
Here,$a_{1} = 111, a_{2} = 107, a_{3} = 103, a_{4} = 99$.
$d_{1} = a_{2} - a_{1} = 107 - 111 = -4$.
$d_{2} = a_{3} - a_{2} = 103 - 107 = -4$.
$d_{3} = a_{4} - a_{3} = 99 - 103 = -4$.
Since the common difference is constant $(d = -4)$,the sequence is an $A.P$.
The $n^{th}$ term formula is $a_{n} = a + (n - 1)d$.
Substituting $a = 111$ and $d = -4$:
$a_{n} = 111 + (n - 1)(-4) = 111 - 4n + 4 = -4n + 115$.
Thus,the $n^{th}$ term is $T_{n} = -4n + 115$.