Unraveling the Mystery of Geometric Sequences and Their Terms

When discussing geometric sequences, understanding how to find the nth term becomes essential. In this article, we delve into a specific problem to showcase the process and highlight the importance of identifying the common ratio and manipulating the given terms.

Introduction to Geometric Sequences

A geometric sequence is a series of numbers in which each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This article will guide you through finding which term of the geometric sequence 2√3, 6√3, ... is equal to 1458.

Identifying the First Term and Common Ratio

The given geometric sequence starts with the term 2√3. The second term is 6√3. To find the common ratio r, we divide the second term by the first term:

r 6√3 / 2√3 6 / 2 3

With the common ratio identified, we can use the general formula for the nth term of a geometric sequence:

an a * rn-1

In this sequence, the first term a is 2√3 and the common ratio r is 3. Thus, the nth term can be written as:

an 2√3 * 3n-1

Finding the Specific Term Equivalent to 1458

To determine which term in the sequence equals 1458, we set up the following equation:

2√3 * 3n-1 1458

We can rearrange this equation to solve for 3n-1:

3n-1 1458 / 2√3

First, we simplify the right-hand side of the equation:

1458 2 * 729 Therefore, 1458 / 2√3 729 / √3

Expressing 729 as a power of 3:

729 36

We can now rewrite the equation:

3n-1 36 / 31/2 311/2

Since the bases are the same, we can set the exponents equal to each other:

n - 1 11 / 2

Solving for n:

n 11 / 2 1 13 / 2 6.5

Since n must be an integer, we examine the terms around 6.5:

For n 6: a6 2√3 * 35 2√3 * 243 486√3 For n 7: a7 2√3 * 36 2√3 * 729 1458√3

Since 1458 is not a term in the sequence, we conclude that there is no integer n such that an 1458.

Conclusion

In geometric sequences where each term is irrational (as in this example), any attempt to match a rational number (like 1458) to a term will not yield an integer value of n. This example demonstrates the necessity of careful manipulation and understanding of the sequence's properties.