For an $A.P.$,the sum of the $4^{th}$ term and the $8^{th}$ term is $24$,while the sum of the $6^{th}$ term and the $10^{th}$ term is $34$. Find the first term $a$ and the common difference $d$ of the $A.P.$

(N/A) The $n^{th}$ term of an $A.P.$ is given by $a_n = a + (n-1)d$.
Given,$a_4 + a_8 = 24$.
$(a + 3d) + (a + 7d) = 24 \implies 2a + 10d = 24 \implies a + 5d = 12$ (Equation $1$).
Also,$a_6 + a_{10} = 34$.
$(a + 5d) + (a + 9d) = 34 \implies 2a + 14d = 34 \implies a + 7d = 17$ (Equation $2$).
Subtracting Equation $1$ from Equation $2$:
$(a + 7d) - (a + 5d) = 17 - 12 \implies 2d = 5 \implies d = 2.5$.
Substituting $d = 2.5$ in Equation $1$:
$a + 5(2.5) = 12 \implies a + 12.5 = 12 \implies a = -0.5$.
Thus,the first term $a = -0.5$ and the common difference $d = 2.5$.