If the mean of first x natural numbers is 26, then find the sum of the first x natural numbers.

Options

A. 1320
B. 1362
C. 1632
D. 1326

Answer

To solve the problem, we need to find the sum of the first \( x \) natural numbers given that their mean is 26. Let's break this down step by step.

### Step 1: Understand the Mean
The mean (average) of the first \( x \) natural numbers can be expressed as:
\[ \text{Mean} = \frac{\text{Sum of first } x \text{ natural numbers}}{x} \]
Given that the mean is 26, we can set up the equation:
\[ \frac{\text{Sum of first } x \text{ natural numbers}}{x} = 26 \]

### Step 2: Formula for the Sum of First \( x \) Natural Numbers
The sum of the first \( x \) natural numbers is given by the formula:
\[ \text{Sum} = \frac{x(x + 1)}{2} \]
Substituting this into our mean equation gives:
\[ \frac{\frac{x(x + 1)}{2}}{x} = 26 \]

### Step 3: Simplify the Equation
We can simplify the left side of the equation:
\[ \frac{x(x + 1)}{2x} = \frac{x + 1}{2} \]
Thus, we have:
\[ \frac{x + 1}{2} = 26 \]

### Step 4: Solve for \( x \)
To eliminate the fraction, we multiply both sides by 2:
\[ x + 1 = 52 \]
Now, subtract 1 from both sides to find \( x \):
\[ x = 52 - 1 = 51 \]

### Step 5: Find the Sum of the First \( x \) Natural Numbers
Now that we have \( x = 51 \), we can find the sum of the first 51 natural numbers using the sum formula:
\[ \text{Sum} = \frac{51(51 + 1)}{2} = \frac{51 \times 52}{2} \]

### Step 6: Calculate the Sum
Calculating this gives:
\[ \text{Sum} = \frac{51 \times 52}{2} = \frac{2652}{2} = 1326 \]

### Final Answer
The sum of the first \( x \) natural numbers is:
\[ \boxed{1326} \]

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