If V is the variance and M is the mean of first 15 natural numbers, then what is `V+M^(2)` equal to

Options

A. 124/3
B. 148/3
C. 248/3
D. 124/9

Answer

To solve the problem, we need to find the variance (V) and the mean (M) of the first 15 natural numbers, and then calculate \( V + M^2 \).

### Step 1: Calculate the Mean (M)

The first 15 natural numbers are:
\[ 1, 2, 3, \ldots, 15 \]

The formula for the mean (M) is given by:
\[ M = \frac{1}{N} \sum_{i=1}^{N} X_i \]
where \( N \) is the number of terms (which is 15 in this case).

The sum of the first \( N \) natural numbers can be calculated using the formula:
\[ \text{Sum} = \frac{N(N + 1)}{2} \]
For \( N = 15 \):
\[ \text{Sum} = \frac{15(15 + 1)}{2} = \frac{15 \times 16}{2} = 120 \]

Now, substituting back into the mean formula:
\[ M = \frac{120}{15} = 8 \]

### Step 2: Calculate the Variance (V)

The formula for variance (V) is given by:
\[ V = \frac{1}{N} \sum_{i=1}^{N} (X_i - M)^2 \]

First, we need to calculate \( (X_i - M)^2 \) for each \( X_i \) from 1 to 15:
- For \( X_1 = 1 \): \( (1 - 8)^2 = 49 \)
- For \( X_2 = 2 \): \( (2 - 8)^2 = 36 \)
- For \( X_3 = 3 \): \( (3 - 8)^2 = 25 \)
- For \( X_4 = 4 \): \( (4 - 8)^2 = 16 \)
- For \( X_5 = 5 \): \( (5 - 8)^2 = 9 \)
- For \( X_6 = 6 \): \( (6 - 8)^2 = 4 \)
- For \( X_7 = 7 \): \( (7 - 8)^2 = 1 \)
- For \( X_8 = 8 \): \( (8 - 8)^2 = 0 \)
- For \( X_9 = 9 \): \( (9 - 8)^2 = 1 \)
- For \( X_{10} = 10 \): \( (10 - 8)^2 = 4 \)
- For \( X_{11} = 11 \): \( (11 - 8)^2 = 9 \)
- For \( X_{12} = 12 \): \( (12 - 8)^2 = 16 \)
- For \( X_{13} = 13 \): \( (13 - 8)^2 = 25 \)
- For \( X_{14} = 14 \): \( (14 - 8)^2 = 36 \)
- For \( X_{15} = 15 \): \( (15 - 8)^2 = 49 \)

Now, we sum these squared differences:
\[ 49 + 36 + 25 + 16 + 9 + 4 + 1 + 0 + 1 + 4 + 9 + 16 + 25 + 36 + 49 = 560 \]

Now, substituting back into the variance formula:
\[ V = \frac{560}{15} = \frac{112}{3} \]

### Step 3: Calculate \( V + M^2 \)

Now we need to calculate \( V + M^2 \):
\[ M^2 = 8^2 = 64 \]
Thus,
\[ V + M^2 = \frac{112}{3} + 64 \]

To add these, we convert 64 into a fraction with a denominator of 3:
\[ 64 = \frac{192}{3} \]
Now, we can add:
\[ V + M^2 = \frac{112}{3} + \frac{192}{3} = \frac{304}{3} \]

### Final Answer

Thus, the final answer is:
\[ \frac{304}{3} \]

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