Question

The value of $\frac{1 \times 2^2 + 2 \times 3^2 + \dots + 100 \times (101)^2}{1^2 \times 2 + 2^2 \times 3 + \dots + 100^2 \times 101}$ is:

• A. $\frac{306}{305}$
• B. $\frac{305}{301}$
• C. $\frac{32}{31}$
• D. $\frac{31}{30}$

Answer 1

Answer: (B)

$\frac{305}{301}$

Answer 2

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:

$\frac{51851000}{25840850} = \frac{305}{301}.$