Understanding the Formula for the Sequence 2, 8, 18, 32, 50

In the digital realm of search engine optimization (SEO), understanding and presenting mathematical sequences can help clarify concepts and enhance content for better engagement. This article delves into the intricacies of identifying and formulating the sequence 2, 8, 18, 32, 50, demonstrating the logical steps and underlying mathematical principles.

Observation and Hypothesis

The sequence in question is 2, 8, 18, 32, 50. At a glance, it might appear random, but a closer look reveals a pattern. Let's start by examining the differences between consecutive terms:

8 - 2 6 18 - 8 10 32 - 18 14 50 - 32 18

The differences are increasing: 6, 10, 14, 18. This suggests an arithmetic progression where the increment itself increases by 4 each time.

Formulating the General Formula

To find a general formula for the sequence, let's observe the pattern more closely:

a1 2
a2 a1 6
a3 a2 10
a4 a3 14
a5 a4 18

From this, we can hypothesize:

an an-1 4n - 2

Now, let's formulate the general formula:

Step-by-Step Derivation

Starting from the first term:

an a1 sum_{k2}^{n}(ak - ak-1)

Since ak - ak-1 4k - 2, we have:

an a1 sum_{k2}^{n}(4k - 2)

Factor out the 4:

an a1 4sum_{k2}^{n}k - 2sum_{k2}^{n}1

Recognizing the sums:

an 2 4left(frac{n(n 1)}{2} - 1right) - 2(n - 1)

Further simplification:

an 2 4left(frac{n^2 n - 2}{2}right) - 2(n - 1) 2 2(n^2 n - 2) - 2(n - 1) 2n^2

Mathematical Analysis Diagram

Verifying the Formula

Let's verify the formula with the given terms:

a1 2 2 × 12 a2 8 2 × 22 a3 18 2 × 32 a4 32 2 × 42 a5 50 2 × 52

Conclusion

In conclusion, the sequence 2, 8, 18, 32, 50 follows the formula an 2n2. While it's easy to hypothesize and verify the formula, it's essential to test predictions with new data. Any discrepancies can lead to refinement or complete rethinking of the hypothesis. Understanding such patterns is crucial in mathematics and can be useful in various applications including SEO and data analysis.

Related Keywords

Sequence formula, arithmetic progression, mathematical rule