Problem 23 - Algebra and Trigonometry

Question

Find the first five terms of the sequence, and determine whether it is geometric. If 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 a common ratio of \(\frac{1}{4}\); the standard form is \(a_n = \frac{1}{4} \left(\frac{1}{4}\right)^{n-1}\).

Step 1: Find the First Term

The first term of the sequence, \(a_1\), is obtained by setting \(n=1\) in the formula given: \(a_1 = \frac{1}{4^1} = \frac{1}{4}\).

Step 2: Find the Second Term

Substitute \(n=2\) into the formula to find the second term: \(a_2 = \frac{1}{4^2} = \frac{1}{16}\).

Step 3: Find the Third Term

For the third term, substitute \(n=3\) into the formula: \(a_3 = \frac{1}{4^3} = \frac{1}{64}\).

Step 4: Find the Fourth Term

Next, substitute \(n=4\) into the formula to find the fourth term: \(a_4 = \frac{1}{4^4} = \frac{1}{256}\).

Step 5: Find the Fifth Term

Finally, substitute \(n=5\) into the formula for the fifth term: \(a_5 = \frac{1}{4^5} = \frac{1}{1024}\).

Step 6: Check if the Sequence is Geometric

A sequence is geometric if each term is a constant multiple of the previous term. Calculate the ratio \(\frac{a_2}{a_1} = \frac{\frac{1}{16}}{\frac{1}{4}} = \frac{1}{4}\), and \(\frac{a_3}{a_2} = \frac{\frac{1}{64}}{\frac{1}{16}} = \frac{1}{4}\). Since the ratio is consistent, it is a geometric sequence.

Step 7: Find the Common Ratio

The common ratio \(r\) is \(\frac{1}{4}\), as calculated in Step 6.

Step 8: Express the nth Term in Standard Form

Since it is a geometric sequence with a first term \(a = \frac{1}{4}\) and a common ratio \(r = \frac{1}{4}\), the \(n\)th term in standard form is: \(a_n = \frac{1}{4} \left(\frac{1}{4}\right)^{n-1}\).

Key Concepts

Common Ratio

In the world of sequences, the common ratio is a key player in defining a geometric sequence. To determine if our sequence is geometric, we need to check if there's a consistent factor, called the common ratio, that relates each term to the previous one. For example, by looking at our given sequence formula \(a_{n} = \frac{1}{4^{n}}\), we find that each term is obtained by dividing the previous one by \(4\). This is seen in calculating \(\frac{a_2}{a_1} = \frac{1}{16} \div \frac{1}{4} = \frac{1}{4}\). The consistency of this division confirms that \(\frac{1}{4}\) is our common ratio. It's crucial because it not only proves the sequence is geometric but also plays a pivotal role in constructing the general formula for the sequence terms. Always remember, finding the common ratio helps you unmask the sequence's hidden geometric nature.

Nth Term Formula

The nth term formula in a geometric sequence is a magic tool that lets you pinpoint any term in the sequence without listing all of them. It follows the structure \(a_n = a r^{n-1}\). Here, \(a\) is the first term, \(r\) is the common ratio, and \(n\) represents the term's position in the sequence. Gained through deep understanding of the concept of sequences, formulating it accurately is essential. In our example, we adapted it to fit our sequence. Since the first term \(a\) is \(\frac{1}{4}\) and common ratio \(r\) is \(\frac{1}{4}\), we write: \(a_n = \frac{1}{4} \left( \frac{1}{4} \right)^{n-1}\). With this formula, you could easily calculate the 10th term or even the 100th without doubling the work—just replace \(n\) with the desired term number. Pretty neat, right?

Sequence Terms

Understanding sequence terms in a geometric sequence is like putting together the pieces of a puzzle. Each term, represented as \(a_n\), is a telltale of the sequence's nature. Our sequence begins with \(n = 1\), providing the first term \(a_1 = \frac{1}{4}\). Continuing with \(n = 2\), \(a_2\) becomes \(\frac{1}{16}\), and so forth. Knowing how these terms relate to each other lets you see the pattern forming. This method of calculation is valuable for confirming patterns as you go, visualizing the entire sequence term by term. With practice, constructing terms won't feel like a daunting task. Instead, it will become a straightforward application of what you already know: the common ratio propels one term to the next, anchored by the first term.