Problem 23 - Algebra and Trigonometry

Question

Find the \(n\)th term of a sequence whose first several terms are given. \(2,4,8,16, \dots\)

Step-by-Step Solution

Answer

The nth term of the sequence is \( a_n = 2^n \).

Step 1: Analyze the Given Sequence

The sequence provided is 2, 4, 8, 16, and so forth. We need to determine how the numbers in the sequence relate to each other. Observe that each number is obtained by multiplying the previous number by 2.

Step 2: Identify the Sequence Type

From the observation that each term is double the previous term, we identify this as a geometric sequence. In a geometric sequence, each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Step 3: Determine the Common Ratio

Calculate the common ratio by dividing any term by its preceding term. For example, divide the second term by the first term: \( \frac{4}{2} = 2 \). Thus, the common ratio \(r\) is 2.

Step 4: Write the General Formula for the Geometric Sequence

The general formula for the nth term of a geometric sequence is given by \( a_n = a \cdot r^{(n-1)} \), where \( a \) is the first term and \( r \) is the common ratio. For this sequence, \( a = 2 \) and \( r = 2 \).

Step 5: Calculate the nth Term Formula

Substitute the values of \( a \) and \( r \) into the general formula: \( a_n = 2 \cdot 2^{(n-1)} \). Simplify: \( a_n = 2^n \). This formula will give the nth term of the sequence.

Key Concepts

Understanding the Common Ratio

In a geometric sequence, the common ratio is a fixed, non-zero number that you multiply each term by to get the next term. It is a central concept needed to analyze and work with geometric sequences. To determine this ratio, you divide a term in the sequence by the previous term. For instance, with the sequence 2, 4, 8, 16, dividing the second term (4) by the first term (2) gives:

• \( \frac{4}{2} = 2 \)

Similarly, dividing the third term (8) by the second term (4) also results in 2. This consistent result confirms the common ratio \( r \) is indeed 2.
Recognizing and calculating this ratio is crucial. It forms the backbone of the sequence and dictates its progression. Identifying the common ratio allows you to apply the correct formula to find any term in the sequence without evaluating every preceding one.

The Nth Term Formula in Geometric Sequences

Once you understand the concept of a common ratio, you can use the nth term formula, which is pivotal in geometric sequence analysis. This formula helps find any term's value in a geometric sequence without listing all the terms. The general formula for the nth term (denoted as \( a_n \)) of a geometric sequence is:

• \( a_n = a \cdot r^{(n-1)} \)

Where:

• \( a \) is the first term of the sequence,
• \( r \) is the common ratio,
• \( n \) is the term number you want to find.

For example, in our sequence, the first term \( a \) is 2, and the common ratio \( r \) is also 2. By substituting these into the formula, we can express the nth term as:

• \( a_n = 2 \cdot 2^{(n-1)} \)

This simplifies to \( a_n = 2^n \). This formula makes it easy to find any term in the sequence quickly by plugging in the term number \( n \).

Sequence Analysis: Patterns in Geometric Sequences

Elsewhere in mathematics, sequence analysis involves examining patterns and structures within sequences to identify forms and behaviors. For geometric sequences, once the common ratio is known, the growth pattern becomes predictable and uniform.
Consider the sequence 2, 4, 8, 16, and so forth. Each term results from multiplying the previous one by 2. Recognizing this pattern is vital for solving problems efficiently and gaining deeper insights into how sequences behave.
One key aspect of sequence analysis is to verify that the sequence maintains its geometric nature throughout. You do this by checking that the ratio between successive terms remains constant. This constancy assures that you have correctly identified a geometric sequence and can confidently apply the nth term formula.
Moreover, analyzing sequences involves predicting future events. For example, if asked to determine the 10th term, using the formula \( a_n = 2^n \), it's simply a matter of substituting and computing:

• \( a_{10} = 2^{10} = 1024 \)

Such analysis is powerful, allowing exploration beyond just the calculated terms, fostering a deeper understanding of geometric progression.