The Secret Behind the Sequence: Unraveling the 6th and 7th Terms

When faced with an unusual sequence of numbers, 1 4 10 20 35, it's natural to wonder what the next terms might be. Initially, one might be tempted to search for patterns or consult resources like OEIS or WolframAlpha. However, in this case, the effort to find patterns proves challenging. This article will explore how to determine the 6th and 7th terms of the sequence and provide insights into the underlying logic.

The Sequence and Its Patterns

Given the sequence 1 4 10 20 35, our goal is to predict the next three terms: the 6th, 7th, and 8th terms. To achieve this, let's first examine the sequence thoroughly and see if we can discern any patterns.

Initial Attempts: Least-Squares Quadratic Fit

When we use WolframAlpha to find a best-fit quadratic formula, it suggests the following equation:

f(x) 1.64286x^2 - 5.15714x 5.8

While this approach might work for finding a smooth curve through the given points, it does not always provide the most accurate insight into the sequence's nature or convey the underlying logic.

Exploring the Hidden Patterns

Let's break down the sequence into different parts to simplify it:

The first sequence of numbers is 1, 4, 10, 20, 35, 56, 84, 120. The second sequence, derived from the first, is 3, 6, 10, 15, 21, 28, 36. The third sequence, derived from the second, is 3, 4, 5, 6, 7, 8.

Upon closer inspection, the second sequence (3, 6, 10, 15, 21, 28, 36) seems to follow a pattern of triangular numbers. For instance, the nth triangular number can be given by the formula:

T(n) n(n 1) / 2

Using this formula, we can predict the next terms in the second sequence:

T(7) 7(7 1) / 2 28, which fits the sequence. T(8) 8(8 1) / 2 36, which also fits the sequence.

The third sequence (3, 4, 5, 6, 7, 8) is simply the sequence of integers starting from 3. This suggests that the next term in the third sequence would be 9.

Understanding the Underlying Formula

The given formula to determine the nth term of the sequence is:

F(n) n(n^2 - n 2) / 6

Let's verify this formula with the given terms:

F(1) 1(1^2 - 1 2) / 6 1 F(2) 2(2^2 - 2 2) / 6 4 F(3) 3(3^2 - 3 2) / 6 10 F(4) 4(4^2 - 4 2) / 6 20 F(5) 5(5^2 - 5 2) / 6 35 F(6) 6(6^2 - 6 2) / 6 56 F(7) 7(7^2 - 7 2) / 6 84 F(8) 8(8^2 - 8 2) / 6 120

Conclusion: Predicting the 6th and 7th Terms

With the formula F(n) n(n^2 - n 2) / 6, we can now predict the 6th and 7th terms of the sequence:

F(6) 56 F(7) 84

The 6th term is 56, and the 7th term is 84. This approach not only helps in determining the next terms but also provides a deeper insight into the nature of the sequence.

Related Keywords

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