Problem 25 - Algebra and Trigonometry

Question

\(25-32\) . 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 given by \(a_n = 2^n\).

Step 1: Identify the Pattern

Look at the list of terms: \(2, 4, 8, 16, \dots \). Notice that each term is a result of multiplying the previous term by \(2\). This suggests a geometric sequence with a common ratio of \(r = 2\).

Step 2: Determine the formula for the n-th term

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

Step 3: Substitute known values into the formula

The first term of this sequence, \(a_1\), is \(2\) and the common ratio \(r\) is \(2\). Substitute these values into the formula: \(a_n = 2 \cdot 2^{n-1}\).

Step 4: Simplify the formula for the n-th term

Combine the constants to simplify: \(a_n = 2^n\). This is the formula for the n-th term of the sequence.

Key Concepts

n-th term formula

In any geometric sequence, where the terms grow according to a specific pattern, the n-th term formula is essential. It helps us find any term's position within a sequence quickly and accurately. Here's the standard n-th term formula for a geometric sequence:

• \( a_n = a_1 \cdot r^{n-1} \)

In this formula:

• \( a_n \) is the n-th term you want to find.
• \( a_1 \) is the first term of the sequence.
• \( r \) represents the common ratio, the fixed number we multiply by to get the next term.
• \( n \) is the term number in the sequence.

Using this formula, you can calculate any term within a geometric sequence as long as you know the first term and the common ratio. This is extremely helpful in solving problems where you need to predict or deduce the behavior of sequences over time. Each term's value is generated by raising the common ratio, \( r \), to the power of one less than the term number (\( n-1 \)), and then multiplying by the first term, \( a_1 \). This formula is both simple and powerful, making it a fundamental tool in sequence analysis.

common ratio

In a geometric sequence, the key to understanding its growth or decline lies in the common ratio. This is the constant factor by which we multiply the current term to get to the next term in the sequence. The common ratio, represented by \( r \), plays a critical role in defining the sequence.To find the common ratio, divide any term in the sequence by the previous term. For example, in the sequence given (\(2, 4, 8, 16, \dots \)), we see:

• \( \frac{4}{2} = 2 \)
• \( \frac{8}{4} = 2 \)
• \( \frac{16}{8} = 2 \)

All these calculations confirm that the common ratio \( r \) is \( 2 \).
Understanding the common ratio is crucial, as it allows us to build the entire sequence from just one term and the ratio itself. If the ratio is greater than 1, the sequence grows; if less than 1 but greater than 0, it shrinks. The overall pattern, whether increasing or decreasing, depends on this common ratio, making it an essential feature of geometric sequences.

sequence patterns

Recognizing sequence patterns is an important skill in mathematics, as it allows you to predict subsequent terms in a sequence, understanding the structure underlying the numbers. In a geometric sequence, the pattern is one where each term results from multiplying the previous term by a consistent number, which is the common ratio.In our example of the sequence \((2, 4, 8, 16, \dots)\), we quickly see a doubling pattern:

• Start at 2, multiply by 2, and get 4.
• From 4, multiply by 2, and reach 8.
• Take 8, multiply by 2, and land on 16.

These steps confirm a uniform, multiplicative increase by the common ratio, producing an exponential growth pattern. Such patterns help not only in identifying the type of sequence but also in establishing an efficient path to writing its formula for any term number \( n \).
Understanding these patterns is vital in different contexts, be it financial growth scenarios or population studies, where similar mathematical principles apply.