Problem 25 - 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 common ratio \(\frac{1}{4}\). The general formula is \(a_n = \frac{1}{4^n}\).

Step 1: Determine the First Five Terms

Substitute the values of \(n = 1, 2, 3, 4, 5\) into the sequence formula \(a_{n} = \frac{1}{4^{n}}\).\- \(a_{1} = \frac{1}{4^{1}} = \frac{1}{4}\)\- \(a_{2} = \frac{1}{4^{2}} = \frac{1}{16}\)\- \(a_{3} = \frac{1}{4^{3}} = \frac{1}{64}\)\- \(a_{4} = \frac{1}{4^{4}} = \frac{1}{256}\)\- \(a_{5} = \frac{1}{4^{5}} = \frac{1}{1024}\)\The first five terms are \(\frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \frac{1}{256}, \frac{1}{1024}\).

Step 2: Check for Geometric Sequence

To determine if the sequence is geometric, check if the ratio between consecutive terms is constant. Calculate \(\frac{a_2}{a_1}, \frac{a_3}{a_2}, \frac{a_4}{a_3}, \frac{a_5}{a_4}\).\\[\frac{a_2}{a_1} = \frac{\frac{1}{16}}{\frac{1}{4}} = \frac{1}{4}\] \\[\frac{a_3}{a_2} = \frac{\frac{1}{64}}{\frac{1}{16}} = \frac{1}{4}\] \\[\frac{a_4}{a_3} = \frac{\frac{1}{256}}{\frac{1}{64}} = \frac{1}{4}\] \\[\frac{a_5}{a_4} = \frac{\frac{1}{1024}}{\frac{1}{256}} = \frac{1}{4}\] \Since the ratio is consistent, the sequence is geometric with a common ratio \(r = \frac{1}{4}\).

Step 3: Determine the General Formula for the Sequence

The standard formula for a geometric sequence is given by \(a_{n} = a \, r^{n-1}\), where \(a\) is the first term, and \(r\) is the common ratio. From our sequence, \(a = \frac{1}{4}\) and \(r = \frac{1}{4}\). Thus, the formula for the \(n\)-th term is:\\[a_n = \frac{1}{4} \left( \frac{1}{4} \right)^{n-1}\] \\[a_n = \frac{1}{4^n}\].

Key Concepts

Common Ratio

In a geometric sequence, one of the defining features is the common ratio. This ratio is a constant value you obtain when you divide any term in the sequence by the preceding term. Consider the sequence from the exercise: \( \frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \frac{1}{256}, \frac{1}{1024} \).

• To find the common ratio, divide the second term by the first term: \( \frac{\frac{1}{16}}{\frac{1}{4}} = \frac{1}{4} \).
• Verify this by dividing the next terms: \( \frac{\frac{1}{64}}{\frac{1}{16}} = \frac{1}{4} \) and so on.
• As you observe, all these calculations yield the same value, \( \frac{1}{4} \), confirming the common ratio remains constant.

When the common ratio is consistent as shown, it signals that the sequence is indeed geometric. This constant element gives geometric sequences their unique property of exponential growth or decay, depending on whether the ratio is greater than or less than one.

Sequence Formula

The sequence formula for a geometric sequence provides a systematic way of finding any term in the sequence without needing to count each one manually. The standard form of a geometric sequence is given by:\[a_{n} = a \cdot r^{n-1}.\]Here, \(a\) represents the initial term, and \(r\) denotes the common ratio. Applying this to the provided sequence:

• We see that \(a = \frac{1}{4} \), which is the first term.
• The common ratio \(r = \frac{1}{4} \).

Therefore, the formula for the \(n\)-th term becomes:\[a_{n} = \frac{1}{4} \cdot \left( \frac{1}{4} \right)^{n-1}.\]This formulation serves as a quick tool to calculate any term in the sequence, emphasizing the power of geometric progression by leveraging the pattern of multiplication and powers. Just plug in any value of \(n\) to find the corresponding term.

Geometric Progression

A geometric progression, often referred to as a geometric sequence, is essentially a set of numbers where each term after the first is derived by multiplying the previous one by a constant, known as the common ratio. These sequences are characterized by their exponential behavior.Some key aspects to understand about geometric progressions include:

• Every number in the sequence is the product of its predecessor and the common ratio.
• This results in a pattern that either increases or decreases quickly, depending on the value of the common ratio.
• If the common ratio is less than 1, like in the given sequence \( \frac{1}{4} \), the terms will get smaller, showcasing exponential decay.

Understanding these core concepts can vastly improve one’s ability to work with sequences in mathematics. The sequence given in the problem is a perfect instance of exponential decay, as the terms successively decreased in size due to the common ratio of \( \frac{1}{4} \). Recognizing these elements aids in a deeper grasp of the powerful nature of geometric sequences.