The sum of three digit numbers which are divisible by 11

Answer

To find the sum of all three-digit numbers that are divisible by 11, we can follow these steps:

### Step 1: Identify the smallest and largest three-digit numbers divisible by 11
The smallest three-digit number divisible by 11 is 110, and the largest is 990.

### Step 2: Recognize that these numbers form an arithmetic progression (AP)
The sequence of three-digit numbers divisible by 11 can be expressed as:
- First term (a) = 110
- Common difference (d) = 11
- Last term (l) = 990

### Step 3: Find the number of terms (n) in this AP
To find the number of terms, we can use the formula for the nth term of an AP:
\[ a_n = a + (n - 1) \cdot d \]
Setting \( a_n = 990 \), we have:
\[ 990 = 110 + (n - 1) \cdot 11 \]

### Step 4: Solve for n
Rearranging the equation:
\[ 990 - 110 = (n - 1) \cdot 11 \]
\[ 880 = (n - 1) \cdot 11 \]
Dividing both sides by 11:
\[ n - 1 = \frac{880}{11} \]
\[ n - 1 = 80 \]
Thus:
\[ n = 80 + 1 = 81 \]

### Step 5: Use the formula for the sum of an AP
The sum \( S_n \) of the first n terms of an AP can be calculated using the formula:
\[ S_n = \frac{n}{2} \cdot (a + l) \]
Substituting the values we found:
\[ S_{81} = \frac{81}{2} \cdot (110 + 990) \]
\[ S_{81} = \frac{81}{2} \cdot 1100 \]

### Step 6: Calculate the sum
Calculating the sum:
\[ S_{81} = \frac{81 \cdot 1100}{2} \]
\[ S_{81} = 81 \cdot 550 \]
\[ S_{81} = 44550 \]

### Final Answer
The sum of all three-digit numbers that are divisible by 11 is **44550**.

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