91Ó°ÊÓ

In any geometric sequence, the common ratio is a crucial component. It determines how each subsequent term is derived from the previous one. To find this ratio, you simply divide the term by the one that precedes it. For instance, in our given sequence, we took the term at position 2, which is \(\frac{1}{27}\), and divided it by the term at position 1, \(\frac{1}{9}\). This calculation gives us the common ratio of \(\frac{1}{3}\).

The common ratio is the same throughout the sequence, which means each term is consistently obtained by multiplying the previous term by this fixed number. If the ratio remains constant, you can confidently use it to find any term within the sequence. Remember, a consistent ratio is what makes the sequence geometric.

Identifying the first term in a geometric sequence is key to determining the entire sequence. The first term is commonly represented by \(a_0\) and serves as the starting point of the whole sequence. Once we have the common ratio, finding the first term becomes straightforward.

In the example sequence, given \(a_1 = \frac{1}{9}\) and the common ratio \(r = \frac{1}{3}\), we can calculate the first term \(a_0\). By rearranging the formula \(a_1 = a_0 \times r\), we solve for \(a_0\) as follows: \(a_0 = \frac{a_1}{r} = \frac{1}{9} \div \frac{1}{3} = \frac{1}{3}\). This starting point, \(\frac{1}{3}\), sets the foundation for the rest of the sequence.

The recursive formula in sequences provides a way to express subsequent terms based on previous ones. For geometric sequences, the recursive formula leverages the common ratio and the prior term

• \(a_{n+1} = a_n \times r\), where \(a_n\) represents the current term.

This formula is particularly powerful because it allows you to construct the entire sequence step by step.

For our sequence, starting with \(a_0 = \frac{1}{3}\) and using the common ratio \(r = \frac{1}{3}\), we apply the recursive formula:

• \(a_1 = a_0 \times r = \frac{1}{3} \times \frac{1}{3} = \frac{1}{9}\)
• \(a_2 = a_1 \times r = \frac{1}{9} \times \frac{1}{3} = \frac{1}{27}\)
• \(a_3 = a_2 \times r = \frac{1}{27} \times \frac{1}{3} = \frac{1}{81}\)

With this formula, determining each next term becomes a simple multiplication task using the common ratio, illustrating the elegance and simplicity of geometric sequences.