A sequence consists of 7 terms arranged in descending order. The mean value of the sequence is 80. If 20 is added to each term and each term is divided by 2 to get a new mean M, then find the difference between M and the original mean.

Points to Remember:

The original sequence has 7 terms in descending order.
The mean of the original sequence is 80.
20 is added to each term, and then each term is divided by 2.
We need to find the difference between the new mean (M) and the original mean (80).

Introduction:

This question involves a simple arithmetic manipulation of a sequence. We are given a sequence of 7 numbers arranged in descending order with a known mean. We are then asked to perform a series of transformations on each term (adding 20 and dividing by 2) and determine the difference between the new mean and the original mean. This requires understanding the effect of linear transformations on the mean of a dataset.

Body:

1. Understanding the Effect of Linear Transformations on the Mean:

Let the original sequence be denoted as {aâ, aâ, aâ, aâ, aâ, aâ, aâ}. The mean of this sequence is given by:

(aâ + aâ + aâ + aâ + aâ + aâ + aâ) / 7 = 80

The sum of the terms in the original sequence is therefore:

Î£aáµ¢ = 80 * 7 = 560

2. Applying the Transformations:

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Each term is modified by adding 20 and then dividing by 2. The new sequence is:

{(aâ + 20)/2, (aâ + 20)/2, (aâ + 20)/2, (aâ + 20)/2, (aâ + 20)/2, (aâ + 20)/2, (aâ + 20)/2}

Let’s find the sum of the terms in the new sequence:

Î£[(aáµ¢ + 20)/2] = (Î£aáµ¢ + Î£20)/2 = (Î£aáµ¢ + 140)/2 = (560 + 140)/2 = 700/2 = 350

3. Calculating the New Mean (M):

The new mean M is the sum of the terms in the new sequence divided by the number of terms (7):

M = 350 / 7 = 50

4. Finding the Difference:

The difference between the new mean (M) and the original mean is:

Difference = M – Original Mean = 50 – 80 = -30

Conclusion:

The original mean of the sequence is 80. After adding 20 to each term and dividing by 2, the new mean (M) becomes 50. The difference between the new mean and the original mean is -30. This demonstrates that adding a constant value to each term in a sequence increases the mean by that constant value, while dividing each term by a constant value divides the mean by that constant value. In this case, the addition of 20 increased the mean by 20, and then dividing by 2 reduced the mean by a factor of 2, resulting in a net decrease of 30 from the original mean. This problem highlights the straightforward impact of linear transformations on statistical measures like the mean, a fundamental concept in descriptive statistics. Understanding these transformations is crucial for data analysis and interpretation across various fields.

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