If the mean of a set of observations `x_(1), x_(2),……,x_(10)` is 40, then the mean of `x_(1)+4, x_(2)+8, x_(3)+12,…….,x_(10)+40` is

Options

A. 54
B. 62
C. 38
D. 50

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 Mean of the Original Set
The mean of the original set of observations \( x_1, x_2, \ldots, x_{10} \) is given as 40. This means:
\[ \text{Mean} = \frac{x_1 + x_2 + \ldots + x_{10}}{10} = 40 \]
Thus, the sum of these observations can be calculated as:
\[ x_1 + x_2 + \ldots + x_{10} = 40 \times 10 = 400 \]

### Step 2: Calculate the New Observations
The new observations are:
- \( x_1 + 4 \)
- \( x_2 + 8 \)
- \( x_3 + 12 \)
- \( x_4 + 16 \)
- \( x_5 + 20 \)
- \( x_6 + 24 \)
- \( x_7 + 28 \)
- \( x_8 + 32 \)
- \( x_9 + 36 \)
- \( x_{10} + 40 \)

### Step 3: Find the Sum of the New Observations
To find the mean of the new set, we first need to calculate the sum of these new observations:
\[ \text{Sum of new observations} = (x_1 + 4) + (x_2 + 8) + (x_3 + 12) + (x_4 + 16) + (x_5 + 20) + (x_6 + 24) + (x_7 + 28) + (x_8 + 32) + (x_9 + 36) + (x_{10} + 40) \]
This can be simplified as:
\[ = (x_1 + x_2 + \ldots + x_{10}) + (4 + 8 + 12 + 16 + 20 + 24 + 28 + 32 + 36 + 40) \]
We already know that \( x_1 + x_2 + \ldots + x_{10} = 400 \).

### Step 4: Calculate the Sum of the Added Constants
Now, we need to calculate the sum of the constants:
\[ 4 + 8 + 12 + 16 + 20 + 24 + 28 + 32 + 36 + 40 \]
This is an arithmetic series where:
- The first term \( a = 4 \)
- The last term \( l = 40 \)
- The number of terms \( n = 10 \)

The sum of an arithmetic series can be calculated using the formula:
\[ S_n = \frac{n}{2} \times (a + l) = \frac{10}{2} \times (4 + 40) = 5 \times 44 = 220 \]

### Step 5: Combine the Sums
Now we can combine the sums:
\[ \text{Total Sum} = 400 + 220 = 620 \]

### Step 6: Calculate the Mean of the New Observations
Finally, we calculate the mean of the new observations:
\[ \text{Mean of new observations} = \frac{\text{Total Sum}}{10} = \frac{620}{10} = 62 \]

### Final Answer
The mean of the new set of observations \( x_1 + 4, x_2 + 8, x_3 + 12, \ldots, x_{10} + 40 \) is \( \boxed{62} \).