Consider the arithmetic sequence \( a_1, a_2, a_3, \ldots, a_n \) where \( a_1 = 1 \) and the common difference \( d = 2 \). The sequence continues such that the \( n \)-th term is \( a_n = 1 + 2(n-1) \). Find the sum of the first 100 terms of this sequence, and then compute the product of this sum with the sum of the first 50 terms of a similar sequence starting at 1 with a common difference of 1. Express your final answer in the form \( \frac{p - \sqrt{q}}{r} \), where \( p, q, \) and \( r \) are integers, and find \( p + q + r \). Let's think step by step and output the final answer within \boxed{}.
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