The $n^{th}$ term of an $A.P.$ is given by $T_{n} = -4n + 15$. Find the $15^{th}$ term and the common difference of the $A.P.$

(A) The $n^{th}$ term of the $A.P.$ is given by $T_{n} = -4n + 15$.
To find the $15^{th}$ term $(T_{15})$,substitute $n = 15$ into the formula:
$T_{15} = -4(15) + 15 = -60 + 15 = -45$.
The common difference $(d)$ of an $A.P.$ given by $T_{n} = an + b$ is the coefficient of $n$.
Here,$d = -4$.
Alternatively,$T_{1} = -4(1) + 15 = 11$ and $T_{2} = -4(2) + 15 = 7$.
$d = T_{2} - T_{1} = 7 - 11 = -4$.