Let $AP(a; d)$ denote the set of all the terms of an infinite arithmetic progression with first term $a$ and common difference $d > 0$. If $AP(1; 3) \cap AP(2; 5) \cap AP(3; 7) = AP(a; d)$,then $a + d$ equals:

If the $p^{th}$ term of an $A.P.$ is $q$ and the $q^{th}$ term is $p$,then its $r^{th}$ term will be:

If the $7^{th}$ term of an $A.P.$ is $40$,then the sum of the first $13$ terms is:

If the sum of the first $n$ even natural numbers is $k$ times the sum of the first $n$ odd natural numbers,then $k = ........$

The sum of the numbers between $100$ and $1000$ which are divisible by $9$ is:

If $\frac{3 + 5 + 7 + \dots + (2n + 1)}{5 + 8 + 11 + \dots + (3n + 2)} = 7$,find the value of $n$. (Note: The original problem implies $n$ terms in the numerator and $10$ terms in the denominator as per the provided solution structure).