Statement-1 :if each of `n` numbers `x_i=i` is replaced by `(i+1)x_i` , then the new mean becomes `((n+1)(5n+4))/6`
Statement-2 : A.M. =`(Sigma_(i=1)^(n) x_i)/n`.

Options

A. a) Both statement I and statement II are correct. Statement II is the correct explanation of statement I.
B. b) Both statement I and statement II are correct. Statement II is not the correct explanation of statement I.
C. c) Statement I is correct and statement II is incorrect.
D. d) Statement I is incorrect and statement II is correct.

Answer

To solve the problem, we need to analyze the statements given and calculate the new mean after replacing the numbers. Let's break down the solution step by step.

### Step 1: Define the original numbers
The original numbers are defined as:
\[ x_i = i \]
for \( i = 1, 2, \ldots, n \).

### Step 2: Replace the original numbers
According to the question, each number \( x_i \) is replaced by:
\[ x_i' = (i + 1)x_i = (i + 1)i = i^2 + i \]

### Step 3: Calculate the new mean
The new mean \( A.M. \) is given by the formula:
\[ A.M. = \frac{\sum_{i=1}^{n} x_i'}{n} \]

Substituting \( x_i' \):
\[ A.M. = \frac{\sum_{i=1}^{n} (i^2 + i)}{n} \]

### Step 4: Separate the summations
We can separate the summation:
\[ A.M. = \frac{\sum_{i=1}^{n} i^2 + \sum_{i=1}^{n} i}{n} \]

### Step 5: Use formulas for summations
We know the formulas for the summations:
1. The sum of the first \( n \) natural numbers:
\[ \sum_{i=1}^{n} i = \frac{n(n + 1)}{2} \]
2. The sum of the squares of the first \( n \) natural numbers:
\[ \sum_{i=1}^{n} i^2 = \frac{n(n + 1)(2n + 1)}{6} \]

### Step 6: Substitute the formulas into the mean
Substituting these formulas into the mean:
\[ A.M. = \frac{\frac{n(n + 1)(2n + 1)}{6} + \frac{n(n + 1)}{2}}{n} \]

### Step 7: Simplify the expression
To simplify, we can factor out \( \frac{n(n + 1)}{6} \):
\[ A.M. = \frac{n(n + 1)}{6n} \left( (2n + 1) + 3 \right) \]
\[ = \frac{n + 1}{6} \left( 2n + 4 \right) \]
\[ = \frac{(n + 1)(2n + 4)}{6} \]

### Step 8: Further simplify
This can be simplified further:
\[ = \frac{(n + 1)(2(n + 2))}{6} = \frac{(n + 1)(n + 2)}{3} \]

### Step 9: Final result
Thus, the new mean is:
\[ A.M. = \frac{(n + 1)(n + 2)}{3} \]

### Conclusion
Now we compare this with the statement given in the question. The statement claims that the new mean is \( \frac{(n + 1)(5n + 4)}{6} \). Since our derived mean does not match this expression, we conclude that Statement-1 is incorrect.