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

The Correct Option is B

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:

\[ \frac{51851000}{25840850} = \frac{305}{301}. \]

Answer 2

Step 1: Express the given sum in mathematical notation.
Given expression:
\[ \frac{1 \times 2^2 + 2 \times 3^2 + \dots + 100 \times (101)^2}{1^2 \times 2 + 2^2 \times 3 + \dots + 100^2 \times 101} \] Or, using sigma notation:
\[ \frac{\sum_{n=1}^{100} n (n+1)^2}{\sum_{n=1}^{100} n^2 (n+1)} \]
Step 2: Expand and simplify numerator and denominator.
Numerator: \[ n(n+1)^2 = n(n^2 + 2n + 1) = n^3 + 2n^2 + n \] Denominator: \[ n^2(n+1) = n^3 + n^2 \] So sums become: \[ \text{Numerator: } \sum_{n=1}^{100} (n^3 + 2n^2 + n) \] \[ \text{Denominator: } \sum_{n=1}^{100} (n^3 + n^2) \]
Step 3: Use sum formulas.
\[ \sum_{n=1}^N n = \frac{N(N+1)}{2} \] \[ \sum_{n=1}^N n^2 = \frac{N(N+1)(2N+1)}{6} \] \[ \sum_{n=1}^N n^3 = \left[ \frac{N(N+1)}{2} \right]^2 \]
Putting \( N = 100 \):
\[ \sum_{n=1}^{100} n = 5050 \] \[ \sum_{n=1}^{100} n^2 = 338350 \] \[ \sum_{n=1}^{100} n^3 = 25502500 \]
Step 4: Calculate the sums.
Numerator: \[ 25502500 + 2 \times 338350 + 5050 = 25502500 + 676700 + 5050 = 26184250 \] Denominator: \[ 25502500 + 338350 = 25840850 \]
Step 5: Compute the required value.
\[ \frac{26184250}{25840850} = \frac{305}{301} \]
Final Answer:
\[ \boxed{\frac{305}{301}} \]