Key Concepts and Formulas

• Arithmetic Progression (AP): A sequence where the difference between consecutive terms is constant (common difference, $d$). The $n$-th term is given by $x_n = x_1 + (n-1)d$.
• Mean of a Sequence: The sum of the terms divided by the number of terms. For $n$ terms, Mean $= \frac{\sum_{i=1}^{n} x_i}{n}$.
• Sum of an AP: The sum of the first $n$ terms of an AP is $S_n = \frac{n}{2}(2x_1 + (n-1)d)$.
• Summation Formulas:
  • Sum of first $n$ natural numbers: $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$
  • Sum of squares of first $n$ natural numbers: $\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$

Step-by-Step Solution

Step 1: Understand the given information and the goal. We are given an arithmetic progression $x_1, x_2, \ldots, x_{100}$ with $x_1 = 2$ and the mean of these 100 terms is 200. We need to find the mean of a new sequence $y_i = i(x_i - i)$ for $1 \leq i \leq 100$.

Step 2: Calculate the sum of the arithmetic progression. The mean of 100 terms is 200. Therefore, the sum of these terms is: $\sum_{i=1}^{100} x_i = \text{Mean} \times \text{Number of terms} = 200 \times 100 = 20000$

Step 3: Find the common difference ($d$) of the AP. The sum of the first 100 terms of an AP can also be expressed as $S_{100} = \frac{100}{2}[2x_1 + (100-1)d]$. We know $x_1 = 2$ and $S_{100} = 20000$. Substituting these values: $20000 = 50[2(2) + 99d]$ $20000 = 50[4 + 99d]$ Divide both sides by 50: $\frac{20000}{50} = 4 + 99d$ $400 = 4 + 99d$ Subtract 4 from both sides: $396 = 99d$ Divide by 99: $d = \frac{396}{99} = 4$ So, the common difference is 4.

Step 4: Express $x_i$ in terms of $i$. Using the formula for the $i$-th term of an AP, $x_i = x_1 + (i-1)d$: $x_i = 2 + (i-1)4$ $x_i = 2 + 4i - 4$ $x_i = 4i - 2$

Step 5: Find the expression for $y_i$. We are given $y_i = i(x_i - i)$. Substitute the expression for $x_i$: $y_i = i((4i - 2) - i)$ $y_i = i(3i - 2)$ $y_i = 3i^2 - 2i$

Step 6: Calculate the sum of $y_i$ for $i=1$ to 100. The sum of $y_i$ is $\sum_{i=1}^{100} y_i = \sum_{i=1}^{100} (3i^2 - 2i)$. We can split this sum using the linearity of summation: $\sum_{i=1}^{100} y_i = 3 \sum_{i=1}^{100} i^2 - 2 \sum_{i=1}^{100} i$ Now, we use the standard summation formulas for $n=100$: $\sum_{i=1}^{100} i = \frac{100(100+1)}{2} = \frac{100 \times 101}{2} = 50 \times 101 = 5050$ $\sum_{i=1}^{100} i^2 = \frac{100(100+1)(2 \times 100 + 1)}{6} = \frac{100 \times 101 \times 201}{6}$ $\sum_{i=1}^{100} i^2 = \frac{2030100}{6} = 338350$ Substitute these values back into the sum of $y_i$: $\sum_{i=1}^{100} y_i = 3(338350) - 2(5050)$ $\sum_{i=1}^{100} y_i = 1015050 - 10100$ $\sum_{i=1}^{100} y_i = 1004950$

Step 7: Calculate the mean of $y_i$. The mean of $y_1, y_2, \ldots, y_{100}$ is $\frac{\sum_{i=1}^{100} y_i}{100}$. $\text{Mean of } y_i = \frac{1004950}{100} = 10049.50$

Common Mistakes & Tips

• Formula Recall: Ensure accurate recall of summation formulas for $i$ and $i^2$. A small error here can lead to a significantly different answer.
• Algebraic Errors: Be meticulous with algebraic manipulations, especially when substituting and simplifying expressions for $x_i$ and $y_i$.
• Calculation Accuracy: Double-check arithmetic, particularly with large numbers in the summation steps. It's easy to misplace a digit.

Summary The problem requires us to first determine the common difference of the given arithmetic progression using its first term and mean. Once the general term $x_i$ is found, we derive the expression for $y_i$. Finally, we calculate the mean of $y_i$ by computing the sum of $y_i$ using standard summation formulas for powers of $i$ and then dividing by the number of terms. This systematic approach leads to the correct mean of the sequence $y_i$.

The final answer is \boxed{10049.50}, which corresponds to option (B).