If the mean of a set of observation `x_1 , x_2 , …….x_10 ` is 20 then the mean of `x_1 +4, x_2 +8, x_3+ 12, ………… x_10 +40 ` is

Options

A. 34
B. 42
C. 38
D. 40

Answer

To find the mean of the new set of observations \( x_1 + 4, x_2 + 8, x_3 + 12, \ldots, x_{10} + 40 \), we can follow these steps:

### Step 1: Understand the Given Information
We know that the mean of the original set of observations \( x_1, x_2, \ldots, x_{10} \) is 20. This means:
\[ \text{Mean} = \frac{x_1 + x_2 + \ldots + x_{10}}{10} = 20 \]
From this, we can find the sum of the observations:
\[ x_1 + x_2 + \ldots + x_{10} = 20 \times 10 = 200 \]

### Step 2: Write the New Observations
The new observations are:
\[ x_1 + 4, x_2 + 8, x_3 + 12, \ldots, x_{10} + 40 \]
We can express this as:
\[ (x_1 + 4) + (x_2 + 8) + (x_3 + 12) + \ldots + (x_{10} + 40) \]

### Step 3: Separate the Original Observations and the Added Constants
We can separate the sum of the original observations from the constants added:
\[ = (x_1 + x_2 + \ldots + x_{10}) + (4 + 8 + 12 + \ldots + 40) \]
We already know that \( x_1 + x_2 + \ldots + x_{10} = 200 \).

### Step 4: Calculate the Sum of the Added Constants
The constants \( 4, 8, 12, \ldots, 40 \) form an arithmetic progression (AP) where:
- First term \( a = 4 \)
- Common difference \( d = 4 \)
- Number of terms \( n = 10 \)

The sum of an AP can be calculated using the formula:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
Substituting the values:
\[ S_{10} = \frac{10}{2} \times (2 \times 4 + (10 - 1) \times 4) \]
\[ = 5 \times (8 + 36) = 5 \times 44 = 220 \]

### Step 5: Calculate the Total Sum of the New Observations
Now we can find the total sum of the new observations:
\[ \text{Total Sum} = 200 + 220 = 420 \]

### Step 6: Find the Mean of the New Observations
Finally, we can find the mean of the new observations:
\[ \text{Mean} = \frac{\text{Total Sum}}{10} = \frac{420}{10} = 42 \]

### Conclusion
The mean of the new set of observations \( x_1 + 4, x_2 + 8, x_3 + 12, \ldots, x_{10} + 40 \) is **42**.

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