Introduction

Understanding the structure and properties of the natural numbers is a fundamental concept in mathematics. In this article, we will explore a specific problem related to the sequence of natural numbers from 1 to 99. Our objective is to find the 50th digit in the sequence formed by writing all natural numbers in order. This task involves breaking down the sequence into understandable segments and applying a systematic approach to identify the desired digit.

Breakdown of the Sequence

To solve the problem of finding the 50th digit in the sequence of natural numbers, we first analyze the contributions of different ranges of numbers to the overall sequence of digits.

Single-Digit Numbers (1 to 9)

Numbers in the range 1 to 9 are single-digit numbers. There are 9 such numbers, and each contributes 1 digit to the sequence.

Total digits contributed by single-digit numbers:

9 digits

Two-Digit Numbers (10 to 99)

The range from 10 to 99 consists of two-digit numbers. We can calculate the number of two-digit numbers as follows:

Number of two-digit numbers 99 - 10 1 90

Since each two-digit number contributes 2 digits, the total number of digits contributed by the two-digit numbers is:

Total digits contributed by two-digit numbers 90 * 2 180 digits

Total Digits from Single-Digit and Two-Digit Numbers

Combining the contributions from both single-digit and two-digit numbers, we get:

Total digits from single-digit and two-digit numbers 9 180 189 digits

Since we are looking for the 50th digit, it falls within the range of two-digit numbers. This is evident because the first 9 digits come from the single-digit numbers, leaving 41 digits to be accounted for from the two-digit numbers.

Identifying the 50th Digit

To find the specific two-digit number that contains the 50th digit, we calculate:

Total digits from two-digit numbers to be covered 41 digits

Number of complete two-digit numbers needed to cover 41 digits:

?41/2? 20 complete two-digit numbers

These 20 two-digit numbers will contribute 20 * 2 40 digits.

The first 20 two-digit numbers are from 10 to 29. Their digits in sequence are:

10 (1 0) 11 (1 1) 12 (1 2) 13 (1 3) 14 (1 4) 15 (1 5) 16 (1 6) 17 (1 7) 18 (1 8) 19 (1 9) 20 (2 0) 21 (2 1) 22 (2 2) 23 (2 3) 24 (2 4) 25 (2 5) 26 (2 6) 27 (2 7) 28 (2 8) 29 (2 9)

The 40th digit is the last digit of 29, which is 9. The 41st digit is the first digit of the next number, which is 30. Hence, the 50th digit in the sequence of natural numbers is 3.

General Approach for Finding the nth Digit

A more generalized approach involves dividing the sequence into ranges and using the division to determine the position of the nth digit. When n is exactly divisible by 20, the digit can be determined by dividing n by 20. For example, for n100, the numerator is 100, the denominator is 20, and when x 100/20, it is an integer 5, indicating the 100th digit is the 5th digit in the 5th set of two-digit numbers.

In the case of n50, 50/20 equals 2.5. Rounding up to the nearest integer gives 3, which is the 50th digit in the sequence.

Conclusion

By systematically breaking down the sequence of natural numbers and identifying the contributions of different ranges, we can precisely locate any specific digit in the sequence. This article demonstrates the step-by-step process for finding the 50th digit. Understanding these techniques is essential for solving similar problems and can be applied in various mathematical and computational contexts.