Unveiling the Pattern of the 8th Term: 2, 6, 18, 54, X

Have you ever stumbled upon a sequence of numbers that seemed so simple yet intriguing? In this article, we will delve into the pattern of the sequence 2, 6, 18, 54, X. By the end of this article, you will not only find out what the next term is in the series, but you will also gain a deeper understanding of geometric sequences and their properties. Let's start by examining the pattern and uncovering the mathematical beauty behind it.

Understanding the Series

The series given is 2, 6, 18, 54, X. A casual glance might not reveal the pattern, but let's look at how each term is generated from the previous one. Notice that each term is a multiple of 3, which hints at a geometric sequence. In a geometric sequence, each term is obtained by multiplying the preceding term by a constant ratio.

Identifying the Common Ratio

The first term (a?) is 2. To find the common ratio (r), we divide the second term by the first term:

r  a? / a?  6 / 2  3

The common ratio is 3, indicating that each term is three times the previous term. This can be represented mathematically using the formula for the nth term of a geometric sequence:

a? a? · r^(n-1)

Calculating the 8th Term

To find the 8th term (a?) of the sequence, we can use the formula above and substitute n 8, a? 2, and r 3:

a?  2 · 3^(8-1)  2 · 3^7

Now, let's compute 3 to the power of 7:

3^7  3 · 3 · 3 · 3 · 3 · 3 · 3  2187

Multiplying this result by 2:

2 · 2187  4374

Hence, the 8th term of the sequence is 4374. This computation aligns with the pattern observed in the series, confirming the geometric nature of the sequence.

Verifying the Pattern

To further verify the pattern, let's write out the multiplication steps to see if the 8th term emerges as 4374:

2 × 3  66 × 3  1818 × 3  5454 × 3  162162 × 3  486486 × 3  14581458 × 3  4374

Each step multiplies the previous term by 3, consistently generating the next term. Therefore, the 8th term is indeed 4374.

Generalizing the Pattern

The pattern can be generalized using the nth term formula for a geometric sequence. For the given series, we can express the nth term as:

a? 2 · 3^(n-1)

Substituting n 8 back into the formula:

a?  2 · 3^(8-1)  2 · 3^7  4374

This confirms our earlier computation and provides a systematic way to find the nth term without explicitly writing out each step.

Conclusion

Through this exploration, we have not only determined the 8th term of the sequence 2, 6, 18, 54, X, but we have also deepened our understanding of geometric sequences and their properties. The sequence can be elegantly represented using a geometric series formula, and the 8th term is a perfect illustration of its power.