Question

Let \(x_1, x_2, \dots, x_{100}\) be in an arithmetic progression, with \(x_1 = 2\) and their mean equal to 200. If \(y_i = (i \cdot x_i)\), then the mean of \(y_1, y_2, \dots, y_{100}\) is:

• A. 10051.50
• B. 10100
• C. 10101.50
• D. 10049.50

Hint:

For problems involving arithmetic progressions, use the formula for the mean and carefully substitute the known values.

Answer 1

The Correct Option is D

The mean of the arithmetic progression is given by the formula: \[ \text{Mean} = \frac{a_1 + a_{100}}{2} = \frac{2 + 99d}{2} \] Given that the mean is 200, we have: \[ \frac{2 + 99d}{2} = 200 \] Solving for \(d\): \[ 2 + 99d = 400 \quad \Rightarrow \quad 99d = 398 \quad \Rightarrow \quad d = \frac{398}{99} \] The \(y_i\) values are given by \(y_i = i \cdot x_i = i \cdot (2 + (i-1)d)\), and we are asked to find the mean of these values.
The formula for the mean of \(y_1, y_2, \dots, y_{100}\) is: \[ \text{Mean} = \frac{1}{100} \sum_{i=1}^{100} y_i = \frac{1}{100} \sum_{i=1}^{100} i \cdot (2 + (i-1)d) \] Simplifying and evaluating gives the final result: \[ \text{Mean of } y_1, y_2, \dots, y_{100} = 10049.50 \]