91Ó°ÊÓ

In a geometric sequence, the first term is the starting point or the initial term from which the sequence begins. It's represented by the letter \( a \). For example, in the sequence \( 2, 2\sqrt{3}, 6, \ldots \), the first term \( a \) is 2. Knowing the first term is crucial because it serves as the foundational value that is multiplied repeatedly by the common ratio to generate subsequent terms. To identify the first term, simply look at the very first number in the sequence.

The common ratio, denoted as \( r \), is the factor by which we multiply each term to obtain the next term in a geometric sequence. To find this ratio, you divide any term in the sequence by the preceding term. For our example sequence \( 2, 2\sqrt{3}, 6, \ldots \), the common ratio is calculated as:
\[ r = \frac{2\sqrt{3}}{2} = \sqrt{3} \]
This common ratio \( \sqrt{3} \) is the multiplier used to generate the next term in the sequence. Identifying the common ratio is essential because it dictates how the sequence progresses. Once you have the common ratio, you can predict all future terms and effectively understand the sequence's behavior.

To find any term in a geometric sequence, we use the nth term formula:
\[ a_n = a \cdot r^{(n-1)} \]
Here, \( a_n \) is the nth term you're looking for, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the position of the term in the sequence.
Suppose we need to find the 9th term in our sequence \( 2, 2\sqrt{3}, 6, \ldots \). Using the formula, we substitute the known values:
\[ a_9 = 2 \cdot (\sqrt{3})^{(9-1)} \]
This simplifies to:
\[ a_9 = 2 \cdot (\sqrt{3})^8 \]
We then simplify further, knowing that \((\sqrt{3})^8 = 81\):
\[ a_9 = 2 \cdot 81 = 162 \]
Therefore, the 9th term of the geometric sequence is 162. By understanding and applying this formula, you can determine any term in the sequence.