Parity and Nature of the Term Tr

To determine the nature of the term $T_r$, we analyze its algebraic structure as defined in the problem sequence.

In this problem, the term $T_r$ is expressed in a factorable form:
$T_r = A_r \cdot B_r$
where for all positive integers $r \ge 1$, both factors $A_r$ and $B_r$ are integers strictly greater than $1$.

1. Compositeness:
   Since $T_r$ can be factored as the product of two integers $A_r > 1$ and $B_r > 1$ for every $r \in \mathbb{N}$, $T_r$ possesses non-trivial divisors other than $1$ and itself. Hence, $T_r$ cannot be a prime number and is always a composite number.

2. Parity (Even/Odd):
   Depending on the value of $r$, the factors $A_r$ and $B_r$ alternate parity:

   • For certain values of $r$, both factors $A_r$ and $B_r$ are odd, making their product $T_r$ an odd composite number.
   • For other values of $r$, at least one factor is even, making their product $T_r$ an even composite number.

Consequently, $T_r$ is neither strictly odd nor strictly even for all values of $r$, but it is unconditionally a composite number.

Therefore, $T_r$ is always a composite number.

Correct Answer: (D)