Which digit is missing in the average of numbers
`9,99,999,9999,….. 999999999?`

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem of finding the missing digit in the average of the numbers \( 9, 99, 999, 9999, \ldots, 999999999 \), we can follow these steps:

### Step 1: Identify the Numbers
The numbers given are:
- \( 9 \) (1 digit)
- \( 99 \) (2 digits)
- \( 999 \) (3 digits)
- \( 9999 \) (4 digits)
- ...
- \( 999999999 \) (9 digits)

### Step 2: Count the Total Numbers
The total numbers in this sequence are from 1 digit to 9 digits. Therefore, there are a total of 9 numbers.

### Step 3: Calculate the Sum of the Numbers
To find the average, we first need to calculate the sum of these numbers:
\[ \text{Sum} = 9 + 99 + 999 + 9999 + 99999 + 999999 + 9999999 + 99999999 + 999999999 \]
This can be simplified as:
\[ \text{Sum} = 9 \times (1 + 11 + 111 + 1111 + 11111 + 111111 + 1111111 + 11111111 + 111111111) \]

### Step 4: Calculate the Sum of the Series
The series \( 1 + 11 + 111 + 1111 + \ldots + 111111111 \) can be calculated. Each term can be expressed as \( \frac{10^n - 1}{9} \) where \( n \) is the number of digits. Thus, we can calculate:
\[ 1 + 11 + 111 + 1111 + 11111 + 111111 + 1111111 + 11111111 + 111111111 = 123456789 \]

### Step 5: Calculate the Total Sum
Now, substituting back into the sum:
\[ \text{Sum} = 9 \times 123456789 = 1111111101 \]

### Step 6: Calculate the Average
To find the average, we divide the total sum by the number of terms:
\[ \text{Average} = \frac{1111111101}{9} = 123456789 \]

### Step 7: Identify the Missing Digit
The average \( 123456789 \) contains the digits \( 1, 2, 3, 4, 5, 6, 7, 8, 9 \). The only digit missing from the set of digits \( 0-9 \) is \( 0 \).

### Final Answer
Thus, the missing digit in the average of the numbers is \( 0 \).

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