Exploring the Enigma of Triangular Numbers: Correct Sequence and Differences

The Triangular Numbers Sequence

When examining the provided sequence 1 4 10 20 35, it is clear that what was meant was the sequence of triangular numbers, not some arbitrary sequence. Triangular numbers are a fascinating sequence in mathematics, where each term represents the sum of the first n natural numbers. Let's delve into the correct sequence and explore the patterns within these numbers.

The Correct Triangular Number Sequence

The sequence of triangular numbers starts with:

1 (the 1st triangular number): 1 3 (the 2nd triangular number): 1 2 3 6 (the 3rd triangular number): 1 2 3 6 10 (the 4th triangular number): 1 2 3 4 10 15 (the 5th triangular number): 1 2 3 4 5 15 21 (the 6th triangular number): 1 2 3 4 5 6 21 28 (the 7th triangular number): 1 2 3 4 5 6 7 28 36 (the 8th triangular number): 1 2 3 4 5 6 7 8 36

The sequence correctly would be: 1, 3, 6, 10, 15, 21, 28, 36, ... instead of the initial sequence given.

The Differences Between Successive Terms

A key characteristic of the triangular numbers is the difference between successive terms. Let's calculate these differences:

3 - 1 2 6 - 3 3 10 - 6 4 15 - 10 5 21 - 15 6 28 - 21 7 36 - 28 8

Explanation of the Differences

The differences between successive triangular numbers follow a pattern as well. Specifically, each difference increases by 1 compared to the previous one. This pattern can be explained by the formula for the nth triangular number:

T_n  n(n   1) / 2

Let's derive the difference between two consecutive triangular numbers:

T_n - T_(n-1)  (n * (n   1) / 2) - (((n - 1) * n) / 2)  n / 2   (n - 1) / 2  n

Thus, the difference between the nth triangular number and the (n-1)th triangular number is always n.

Conclusion

In conclusion, the sequence of triangular numbers is 1, 3, 6, 10, 15, 21, 28, 36, ... and the differences between successive terms increase by 1 each time. This pattern and the nature of triangular numbers make them a fascinating subject for study in mathematics and data analysis.