Problem 23 - Precalculus: Mathematical for Calculus

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 n-th term is \(2^n\).

Step 1: Identify the Pattern

The sequence given is: 2, 4, 8, 16, ... To find the pattern, observe the relationship between the consecutive terms. Each term is obtained by multiplying the previous term by 2. This indicates a geometric sequence with a common ratio.

Step 2: Determine the Formula for the n-th Term

In a geometric sequence, the n-th term is given by the formula \(a_n = a_1 imes r^{n-1}\), where \(a_1\) is the first term and \(r\) is the common ratio. Here, \(a_1 = 2\) and \(r = 2\).

Step 3: Write the n-th Term Formula

By substituting the values of \(a_1\) and \(r\) into the formula from Step 2, the n-th term of the sequence is \(a_n = 2 \times 2^{n-1}\).

Step 4: Simplify the Formula

The formula for the n-th term can be simplified further: \(a_n = 2^n\). Thus, every term in the sequence is \(2^n\), where \(n\) is the position of the term in the sequence.

Key Concepts

Common Ratio

In a geometric sequence, each term is generated by multiplying the previous term by a constant value known as the "common ratio". Being able to recognize and accurately determine this common ratio is crucial for understanding the sequence's behavior and predicting subsequent terms.
For instance, in our given sequence (2, 4, 8, 16), you can find the common ratio by dividing any term by the preceding one, such as 4 divided by 2, 8 divided by 4, and so on. In each case, the result is 2. This consistent value of 2 is our common ratio. This means that each term in the sequence is double the one before it.

• Observation: Recognizing this common ratio allows you to predict future terms by simply multiplying by 2.
• Significance: The common ratio dictates the rate at which the sequence grows.

Understanding this concept not only helps in solving problems but also in grasping the dynamic nature of geometric sequences.

n-th Term Formula

The n-th term formula in a geometric sequence allows you to find any term in the sequence without manually calculating all preceding terms. This is done by using the first term and the common ratio, which means you don't always need the entire sequence to find any given term.
A standard n-th term formula for a geometric sequence is given by: \[ a_n = a_1 \times r^{n-1} \] where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term position in the sequence.
In our example, the first term \( a_1 \) is 2, and the common ratio \( r \) is also 2. Therefore, the formula applies as:

• \( a_n = 2 \times 2^{n-1} \)
• By simplifying it, we get \( a_n = 2^n \).

The power \( n-1 \) accounts for the number of times you have to multiply by the common ratio, starting from the first term. This provides a direct way to calculate any term's value without building the sequence term by term.

Pattern Recognition

Pattern recognition is a key skill in identifying sequences such as geometric ones. By recognizing patterns, you can establish relationships between terms that help you formulate rules or formulas. Looking at our sequence (2, 4, 8, 16), observe how each term is related to another through the operation of multiplication. This gives a clear indication that the sequence is geometric.
Here's how you can approach pattern recognition in sequences:

• Identify Patterns Easily: Look for a repeated process, such as multiplying or adding, that can describe how each term is formed from the previous term.
• Use Patterns to Predict: Once you see a pattern, you can forecast future terms, helping you solve problems more swiftly.
• Apply Patterns to Find Formulas: Identifying the pattern helps in formulating a general expression, such as the n-th term formula, which succinctly represents the sequence.

Mastering pattern recognition simplifies complex problems, enabling you to quickly determine the formula that governs the sequence without having to compute each term manually.