Statement-1 :The A.M. of series 1,2,4,8,16,…… `2^n` is `(2^(n+1)-1)/(n+1)`
Statement-2 : Arithmetic mean (A.M.) of ungrouped data is `(Sigmax_i)/n` where `x_1,x_2`…. `x_n` are n numbers .

Options

A. (a)Statement-1 is True, Statement-2 is True , Statement-2 is a correct explanation for Statement-1
B. (b)Statement-1 is True, Statement-2 is True , Statement-2 is NOT a correct explanation for Statement-1
C. (c)Statement-1 is True , Statement-2 is False
D. (d)Statement-1 is False , Statement-2 is True

Answer

To solve the question regarding the two statements about the Arithmetic Mean (A.M.) of a series and ungrouped data, we will analyze each statement step by step.

### Step 1: Understanding Statement 1
**Statement 1:** The A.M. of the series \(1, 2, 4, 8, 16, \ldots, 2^n\) is given by \(\frac{2^{n+1} - 1}{n + 1}\).

1. **Identify the Series:** The series is a geometric progression where the first term \(a = 1\) (which is \(2^0\)) and the common ratio \(r = 2\).
2. **Number of Terms:** The number of terms in the series from \(2^0\) to \(2^n\) is \(n + 1\) (since it includes \(2^0\) to \(2^n\)).
3. **Sum of the Series:** The sum \(S_n\) of the first \(n + 1\) terms of a geometric series can be calculated using the formula:
\[ S_n = a \frac{r^{n+1} - 1}{r - 1} \]
Substituting \(a = 1\) and \(r = 2\):
\[ S_n = 1 \cdot \frac{2^{n+1} - 1}{2 - 1} = 2^{n+1} - 1 \]

### Step 2: Calculate the Arithmetic Mean
4. **Arithmetic Mean Calculation:** The A.M. is given by the formula:
\[ A.M. = \frac{S_n}{\text{Number of Terms}} = \frac{2^{n+1} - 1}{n + 1} \]
This confirms that Statement 1 is true.

### Step 3: Understanding Statement 2
**Statement 2:** The A.M. of ungrouped data is given by \(\frac{\Sigma x_i}{n}\), where \(x_1, x_2, \ldots, x_n\) are \(n\) numbers.

5. **Definition of A.M.:** The arithmetic mean of a set of \(n\) numbers \(x_1, x_2, \ldots, x_n\) is calculated as:
\[ A.M. = \frac{x_1 + x_2 + \ldots + x_n}{n} = \frac{\Sigma x_i}{n} \]
This confirms that Statement 2 is also true.

### Conclusion
Both statements are true:
- Statement 1 is true as we derived the A.M. of the geometric series correctly.
- Statement 2 is true as it correctly defines the A.M. for ungrouped data.

### Final Answer
Both statements are true, and Statement 2 is a correct explanation for Statement 1.

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