91Ó°ÊÓ

In a geometric sequence, the common ratio is the factor by which you multiply each term to get the next one. It remains constant throughout the sequence. This characteristic is what defines a geometric sequence. To find the common ratio, you take any term in the sequence and divide it by the preceding term.

For example, in the provided sequence, you start with the given terms 32 and 128. Compute the common ratio as follows:

• Divide the third term by the second term: \( r = \frac{128}{32} \)
• This simplifies to \( r = 4 \)

Double-checking this ratio involves ensuring that when multiplying the second term, 32, by the common ratio of 4, you correctly achieve the third term, 128. This consistent multiplier assures you that you are indeed working with a geometric sequence.

Identifying the first term of a geometric sequence sets the stage for understanding the entire pattern. This term, often denoted as \( a_1 \), is the starting point from which all other terms of the sequence are generated using the common ratio.

In the given sequence, finding the first term is straightforward once you have determined the common ratio. Given that the second term is 32 and the common ratio is 4, you can find the first term \( a_1 \) by reversing the multiplication. You do this by dividing the second term by the common ratio:

• \( a_1 = \frac{32}{4} \)
• Thus, \( a_1 = 8 \)

Knowing \( a_1 = 8 \) means the sequence starts at this number, providing you a reference point for applying the sequence rule to find any other term.

The sequence rule in a geometric sequence provides a formula to calculate any term's value based on its position. The general form of the rule is given by:

• \( a_n = a_1 \times r^{(n-1)} \)

Where \( a_n \) is the term you're trying to find, \( a_1 \) is the first term, and \( r \) is the common ratio.

For the exercise in question, with a common ratio \( r = 4 \) and a first term \( a_1 = 8 \), the sequence rule becomes:

• \( h(n) = 8 \times 4^{(n-1)} \)

This rule now empowers you to insert any integer value for \( n \) to calculate the nth term of the sequence effectively, making it a powerful tool for both analysis and prediction of the sequence's behavior.