91Ó°ÊÓ

In any geometric sequence, the term "common ratio" is key to understanding its structure. The common ratio, often denoted as \( r \), is the factor by which we multiply each term to get to the next term. For a sequence to be geometric, this ratio must remain constant between consecutive terms.

In our exercise, we determined the common ratio by examining the first two terms of the sequence. We took the second term \( a_2 = \frac{1}{16} \) and divided it by the first term \( a_1 = \frac{1}{4} \):

• \( r = \frac{a_2}{a_1} = \frac{\frac{1}{16}}{\frac{1}{4}} = \frac{1}{4} \)

Repeating this process with subsequent terms ensures that the ratio remains the same, confirming the sequence's geometric nature. This verification is crucial as it validates our approach to defining the sequence formula.

Remember, a consistent common ratio is what distinguishes a geometric sequence from other sequences, such as arithmetic sequences where we add or subtract a constant instead.

The sequence formula is our starting point in understanding sequences, including geometric ones. It provides a rule or pattern that describes a sequence's terms. In general, a geometric sequence can be expressed using the formula \( a_n = ar^{n-1} \), where:

• \( a \) is the first term of the sequence
• \( r \) stands for the common ratio
• \( n \) represents the position of the term in the sequence

In the given exercise, the sequence formula provided was \( a_n = \frac{1}{4^n} \). This already hints at the usage of powers, typical for geometric sequences.

The formula gives quick access to any term in the sequence without doing all multiplication steps from the first term, which is particularly useful for terms further down the line. Not only does it save time, but it also minimizes errors in calculation.

Finding the \( n \)th term in a geometric sequence is straightforward once you have the sequence formula. This term depends on its position in the sequence and can be effortlessly calculated using the formula \( a_n = ar^{n-1} \). This form ties well into our exercise, where the sequence formula was given as \( a_n = \frac{1}{4^n} \).

Using the general form, let's break it down:

• The first term \( a = \frac{1}{4} \)
• The common ratio \( r = \frac{1}{4} \)

So, the nth term is expressed as:

• \( a_n = \frac{1}{4}(\frac{1}{4})^{n-1} \)

This simplifies to \( a_n = \frac{1}{4^n} \), exactly what was provided in the exercise. Having the ability to express any term with this formula underscores the predictability and uniformity inherent in geometric sequences. This is an invaluable tool for students trying to decode complex problems involving sequences.