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:
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:
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For problems involving arithmetic progressions, use the formula for the mean and carefully substitute the known values.
Updated On: Mar 25, 2026
- 10051.50
- 10100
- 10101.50
13433
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The Correct Option is D
Solution and Explanation
We are given:
\(x_1, x_2, \dots, x_{100}\) are in an arithmetic progression (A.P.)
\(x_1 = 2\)
Mean of all terms = 200
Step 1: Find the last term \(x_{100}\).
For an A.P., mean of all terms is:
\[ \text{Mean} = \frac{x_1 + x_{100}}{2} \] Given mean = 200:
\[ 200 = \frac{2 + x_{100}}{2} \] \[ 400 = 2 + x_{100} \] \[ x_{100} = 398 \]
Step 2: Find the common difference \(d\).
\[ x_{100} = x_1 + 99d \] \[ 398 = 2 + 99d \] \[ 396 = 99d \] \[ d = 4 \]
So,
\[ x_i = 2 + (i-1)4 = 4i - 2 \]
Step 3: Define \(y_i\).
Given:
\[ y_i = i \cdot x_i = i(4i - 2) = 4i^2 - 2i \]
Step 4: Find the mean of \(y_1, y_2, \dots, y_{100}\).
Mean =
\[ \frac{1}{100}\sum_{i=1}^{100} (4i^2 - 2i) \] \[ = \frac{1}{100} \left(4\sum i^2 - 2\sum i \right) \]
Use standard formulas:
\[ \sum_{i=1}^{100} i = \frac{100 \cdot 101}{2} = 5050 \] \[ \sum_{i=1}^{100} i^2 = \frac{100 \cdot 101 \cdot 201}{6} = 338350 \]
Substitute:
\[ 4\sum i^2 = 4 \cdot 338350 = 1353400 \] \[ 2\sum i = 2 \cdot 5050 = 10100 \]
So,
\[ \sum y_i = 1353400 - 10100 = 1343300 \]
Mean =
\[ \frac{1343300}{100} = 13433 \]
Final Answer:
\[ \boxed{13433} \]
\(x_1, x_2, \dots, x_{100}\) are in an arithmetic progression (A.P.)
\(x_1 = 2\)
Mean of all terms = 200
Step 1: Find the last term \(x_{100}\).
For an A.P., mean of all terms is:
\[ \text{Mean} = \frac{x_1 + x_{100}}{2} \] Given mean = 200:
\[ 200 = \frac{2 + x_{100}}{2} \] \[ 400 = 2 + x_{100} \] \[ x_{100} = 398 \]
Step 2: Find the common difference \(d\).
\[ x_{100} = x_1 + 99d \] \[ 398 = 2 + 99d \] \[ 396 = 99d \] \[ d = 4 \]
So,
\[ x_i = 2 + (i-1)4 = 4i - 2 \]
Step 3: Define \(y_i\).
Given:
\[ y_i = i \cdot x_i = i(4i - 2) = 4i^2 - 2i \]
Step 4: Find the mean of \(y_1, y_2, \dots, y_{100}\).
Mean =
\[ \frac{1}{100}\sum_{i=1}^{100} (4i^2 - 2i) \] \[ = \frac{1}{100} \left(4\sum i^2 - 2\sum i \right) \]
Use standard formulas:
\[ \sum_{i=1}^{100} i = \frac{100 \cdot 101}{2} = 5050 \] \[ \sum_{i=1}^{100} i^2 = \frac{100 \cdot 101 \cdot 201}{6} = 338350 \]
Substitute:
\[ 4\sum i^2 = 4 \cdot 338350 = 1353400 \] \[ 2\sum i = 2 \cdot 5050 = 10100 \]
So,
\[ \sum y_i = 1353400 - 10100 = 1343300 \]
Mean =
\[ \frac{1343300}{100} = 13433 \]
Final Answer:
\[ \boxed{13433} \]
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