What is the average of all the natural numbers from 49 to 125 ?

Options

A. 85
B. 87
C. 88
D. 86

Answer

To find the average of all the natural numbers from 49 to 125, we can follow these steps:

### Step 1: Identify the range of numbers
We are looking for the average of natural numbers starting from 49 to 125.

### Step 2: Calculate the sum of natural numbers from 1 to 125
We use the formula for the sum of the first n natural numbers:
\[ \text{Sum} = \frac{n(n + 1)}{2} \]
For \( n = 125 \):
\[ \text{Sum}_{1 \text{ to } 125} = \frac{125 \times 126}{2} = \frac{15750}{2} = 7875 \]

### Step 3: Calculate the sum of natural numbers from 1 to 48
Using the same formula for \( n = 48 \):
\[ \text{Sum}_{1 \text{ to } 48} = \frac{48 \times 49}{2} = \frac{2352}{2} = 1176 \]

### Step 4: Calculate the sum of natural numbers from 49 to 125
Now, we subtract the sum of the first 48 numbers from the sum of the first 125 numbers:
\[ \text{Sum}_{49 \text{ to } 125} = \text{Sum}_{1 \text{ to } 125} - \text{Sum}_{1 \text{ to } 48} = 7875 - 1176 = 6699 \]

### Step 5: Calculate the number of natural numbers from 49 to 125
To find the number of natural numbers in this range:
\[ \text{Number of terms} = 125 - 49 + 1 = 77 \]

### Step 6: Calculate the average
The average is given by the formula:
\[ \text{Average} = \frac{\text{Sum}}{\text{Number of terms}} = \frac{6699}{77} \]
Calculating this gives:
\[ \text{Average} = 87 \]

### Final Answer
Thus, the average of all the natural numbers from 49 to 125 is **87**.

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