1,2,3,4,10,11,12,13,14,20,21,22,23,24,… are the consecutive numbes written in base 5. The twenty fifth number in the above sequence would be :

Options

A. 52
B. 100
C. 25
D. none of these

Answer

To find the 25th number in the sequence of consecutive numbers written in base 5, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Base 5 System**:
- In base 5, the digits used are 0, 1, 2, 3, and 4. After 4, the next number is represented as 10 (which is equivalent to 5 in decimal).

2. **List the Sequence**:
- The given sequence starts with: 1, 2, 3, 4, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24.
- We can see that the numbers from 1 to 4 are single-digit numbers in base 5.
- The numbers from 5 (which is 10 in base 5) to 24 (which is 44 in base 5) are two-digit numbers.

3. **Count the Numbers**:
- The numbers from 1 to 4 give us 4 numbers.
- The numbers from 5 (10 in base 5) to 24 (44 in base 5) give us:
- 10 (5 in decimal) to 14 (9 in decimal) = 5 numbers.
- 20 (10 in decimal) to 24 (14 in decimal) = 5 numbers.
- So far, we have:
- 4 (from 1 to 4) + 5 (from 10 to 14) + 5 (from 20 to 24) = 14 numbers.

4. **Continue Counting**:
- The next set of numbers starts from 25 (which is 100 in base 5).
- The numbers from 25 (100 in base 5) to 44 (which is 24 in decimal) will be:
- 100, 101, 102, 103, 104 (5 numbers).
- Now we have:
- 14 (previous total) + 5 (from 100 to 104) = 19 numbers.

5. **Next Set of Numbers**:
- The next set starts from 30 (which is 15 in decimal) to 34 (which is 19 in decimal):
- 110, 111, 112, 113, 114 (5 numbers).
- Now we have:
- 19 (previous total) + 5 (from 110 to 114) = 24 numbers.

6. **Finding the 25th Number**:
- The next number in the sequence after 114 (which is 24 in decimal) is 120 (which is 25 in decimal).
- Thus, the 25th number in the sequence is 120 in base 5.

### Final Answer:
The 25th number in the given sequence is **50** in base 5.