Problem 23 - Precalculus Mathematics for Calculus

Question

Find the first five terms of the sequence, and determine whether it is geometric. If it is geometric, find the common ratio, and express the \(n\) th term of the sequence in the standard form \(a_{n}=a r^{n-1}\). $$a_{n}=\frac{1}{4^{n}}$$

Step-by-Step Solution

Answer

The sequence is geometric with first five terms \(\frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \frac{1}{256}, \frac{1}{1024}\). The common ratio is \(\frac{1}{4}\), and the nth term is \(a_n=\frac{1}{4^n}\).

Step 1: Identify the Sequence Formula

The given sequence formula is \( a_n = \frac{1}{4^n} \). We need to use this formula to find the first five terms of the sequence.

Step 2: Calculate the First Term

Substitute \( n = 1 \) into the formula to find the first term of the sequence:\[ a_1 = \frac{1}{4^1} = \frac{1}{4} \]

Step 3: Calculate the Second Term

Substitute \( n = 2 \) into the formula:\[ a_2 = \frac{1}{4^2} = \frac{1}{16} \]

Step 4: Calculate the Third Term

Substitute \( n = 3 \) into the formula:\[ a_3 = \frac{1}{4^3} = \frac{1}{64} \]

Step 5: Calculate the Fourth Term

Substitute \( n = 4 \) into the formula:\[ a_4 = \frac{1}{4^4} = \frac{1}{256} \]

Step 6: Calculate the Fifth Term

Substitute \( n = 5 \) into the formula:\[ a_5 = \frac{1}{4^5} = \frac{1}{1024} \]

Step 7: Determine if the Sequence is Geometric

A sequence is geometric if the ratio between consecutive terms is constant. Calculate the ratio \( r \) between the first two terms:\[ r = \frac{a_2}{a_1} = \frac{\frac{1}{16}}{\frac{1}{4}} = \frac{1}{4} \]Check if \( r \) remains the same for subsequent terms:\[ \frac{a_3}{a_2} = \frac{\frac{1}{64}}{\frac{1}{16}} = \frac{1}{4}, \quad \frac{a_4}{a_3} = \frac{1}{4}, \quad \frac{a_5}{a_4} = \frac{1}{4} \]The ratio is constant, so the sequence is geometric.

Step 8: Write the General Formula

Since the sequence is geometric with a common ratio \( r = \frac{1}{4} \) and the first term is \( a = \frac{1}{4} \), we express the nth term in the form \( a_n = ar^{n-1} \):\[ a_n = \frac{1}{4} \left(\frac{1}{4}\right)^{n-1} = \frac{1}{4^n} \]

Key Concepts

Common Ratio

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.

Sequence Formula

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.

Nth Term

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.