We are asked to evaluate the expression:
\[ \frac{1 \times 2^2 + 2 \times 3^2 + \cdots + 100 \times (101)^2}{1^2 \times 2^2 + 2^2 \times 3^2 + \cdots + 100^2 \times 101}. \]
This can be rewritten as:
\[ \frac{\sum_{r=1}^{100} r(r+1)^2}{\sum_{r=1}^{100} r^2(r+1)}. \]
Now, expand both the numerator and denominator:
Numerator: \[ \sum_{r=1}^{100} r(r+1)^2 = \sum_{r=1}^{100} r(r^2 + 2r + 1) = \sum_{r=1}^{100} (r^3 + 2r^2 + r). \] Denominator: \[ \sum_{r=1}^{100} r^2(r+1) = \sum_{r=1}^{100} (r^3 + r^2). \]
We now need to compute these sums:
\[ \sum_{r=1}^{100} r^3 = \left(\frac{100(100+1)}{2}\right)^2 = 25502500. \] \[ \sum_{r=1}^{100} r^2 = \frac{100(100+1)(2 \times 100+1)}{6} = 338350. \]
Using these values, we can calculate:
Numerator: \[ 25502500 + 2 \times 338350 + 5050 = 51851000. \] Denominator: \[ 25502500 + 338350 = 25840850. \]
Thus, the value of the expression is:
\[ \frac{51851000}{25840850} = \frac{305}{301}. \]