Discovering the Pattern in the Sequence: 2, 6, 12, 20, 30, 42, 56, 72, ...

Sequences are a fascinating topic in mathematics, and one such intriguing sequence is 2, 6, 12, 20, 30, 42, 56, 72, ... This sequence holds a hidden pattern that, when uncovered, provides a clear understanding of its next term and extends the knowledge of mathematical sequences. In this article, we will dive deep into the pattern and identify the next number in the sequence, as well as provide a general formula to predict any term in the sequence.

Identifying the Pattern

Step-by-Step Analysis

The given sequence is:

2, 6, 12, 20, 30, 42, 56, 72, ...

Upon closer examination, we can see that each term is generated by adding a specific pattern to the previous term.

Calculating the Differences

Let's calculate the differences between consecutive terms to understand the pattern:

6 - 2 4 12 - 6 6 20 - 12 8 30 - 20 10 42 - 30 12 56 - 42 14 72 - 56 16

We notice that the differences form a pattern of even numbers: 4, 6, 8, 10, 12, 14, 16, ... This pattern suggests that each difference increases by 2. So, the next difference should be 18 (16 2).

Therefore, the next term in the sequence can be found by adding 18 to the last term (72):

72 18 90

Hence, the next number in the sequence is 90.

Exploring the Quadratic Sequence Formula

The sequence can also be represented using a quadratic formula. Let's explore this further:

We can rewrite the sequence as:

2 1 × 2 6 2 × 3 12 3 × 4 20 4 × 5 30 5 × 6 42 6 × 7 56 7 × 8 72 8 × 9

From this, we can deduce that the nth term can be represented as:

T_n n times (n 1)

Using this formula, we can calculate the next term in the sequence:

T_9 9 times 10 90

Pattern Analysis and General Formula

The pattern in the differences (4, 6, 8, 10, 12, 14, 16, ...) can also be analyzed in terms of a quadratic sequence. The differences form an arithmetic sequence with the first term as 4 and a common difference of 2.

The general formula for an arithmetic sequence is:

a_n a_1 (n - 1)d

Where:

a1 4 (the first term) d 2 (the common difference)

The nth term of the differences can be calculated as:

d_n 4 (n - 1) × 2

The next difference will be:

d_8 4 (8 - 1) × 2 18

Thus, the next term in the sequence is 72 18 90.

Conclusion

By exploring the pattern in the sequence 2, 6, 12, 20, 30, 42, 56, 72, we have uncovered that the next term is 90. This sequence follows a clear pattern that can be generalized:

Step-by-step calculation suggests adding 2n to the previous term for the next term. Quadratic formula Tn n × (n 1) provides a direct way to find any term in the sequence.

This exploration not only helps in identifying the next term but also deepens our understanding of sequences and patterns in mathematics.