Problem 36 - Precalculus: Mathematics for Calculus

Question

Determine the common ratio, the fifth term, and the \(n\) th term of the geometric sequence. $$t, \frac{t^{2}}{2}, \frac{t^{3}}{4}, \frac{t^{4}}{8}, \dots$$

Step-by-Step Solution

Answer

The common ratio is \( \frac{t}{2} \); the fifth term is \( \frac{t^5}{16} \); the \( n \) th term is \( \frac{t^n}{2^{n-1}} \).

Step 1: Identify the First Term

The first term of the sequence is given as \( t \). It is denoted by \( a \). So, \( a = t \).

Step 2: Determine the Common Ratio

In a geometric sequence, each term is the previous term multiplied by the common ratio \( r \). To find \( r \), divide the second term by the first term: \[ r = \frac{\frac{t^2}{2}}{t} = \frac{t}{2}. \] Thus, the common ratio \( r \) is \( \frac{t}{2} \).

Step 3: Find the Fifth Term

The formula for the \( n \) th term of a geometric sequence is \( a_n = a \times r^{n-1} \). To find the fifth term \( a_5 \): \[ a_5 = t \times \left( \frac{t}{2} \right)^{4} = t \times \frac{t^4}{16} = \frac{t^5}{16}. \] Thus, the fifth term is \( \frac{t^5}{16} \).

Step 4: Express the nth Term

Using the same formula for the \( n \) th term, \( a_n = a \times r^{n-1} \), substitute \( a = t \) and \( r = \frac{t}{2} \): \[ a_n = t \times \left( \frac{t}{2} \right)^{n-1} = \frac{t^n}{2^{n-1}}. \] Therefore, the \( n \) th term is \( \frac{t^n}{2^{n-1}} \).

Key Concepts

Understanding the Common Ratio in Geometric Sequences

The common ratio in a geometric sequence is an essential concept that will help you understand how each term relates to the ones before and after it. In simple terms, the common ratio is the factor by which one term of the sequence is multiplied to get the next term. To find the common ratio, divide any term in the sequence by the term that comes directly before it.
For example, if you have a sequence like this one:

• First term: \( t \)
• Second term: \( \frac{t^2}{2} \)
• Third term: \( \frac{t^3}{4} \)

The common ratio \( r \) is found by dividing the second term by the first term: \( r = \frac{\frac{t^2}{2}}{t} = \frac{t}{2} \). If you apply this process to each pair of consecutive terms in the sequence, you will see the same ratio, \( \frac{t}{2} \). This tells you that you multiply each term by \( \frac{t}{2} \) to get the subsequent term.

How to Determine the nth Term of a Geometric Sequence

Determining the \( n \)th term of a geometric sequence involves using a specific formula. This formula allows you to find any term in the sequence without calculating all the preceding terms. The general formula for the \( n \)th term \( a_n \) of a geometric sequence is:

• \( a_n = a \times r^{n-1} \)

Where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the term number you wish to find.
In our example, the first term \( a \) is \( t \), and the common ratio \( r \) is \( \frac{t}{2} \). Plugging these values into the formula gives you:

• \( a_n = t \times \left( \frac{t}{2} \right)^{n-1} \)

By simplifying, you find that the \( n \)th term is \( \frac{t^n}{2^{n-1}} \). This formula efficiently calculates any term in the sequence, even those with large indices.

Exploring Geometric Series

A geometric series is the sum of the terms of a geometric sequence. While the sequence provides the individual terms, the series combines them through addition. In essence, if you have a geometric sequence, the geometric series is what you get when you sum together a specific number of terms.
For example, if the geometric sequence is \( t, \frac{t^2}{2}, \frac{t^3}{4}, \ldots \), then the sum of the first four terms is the geometric series:

• \( t + \frac{t^2}{2} + \frac{t^3}{4} + \frac{t^4}{8} \)

Calculating a geometric series often involves formulas that account for infinite or finite numbers of terms. For a finite series, the sum \( S_n \) can be calculated using the formula:

• \( S_n = a \frac{1-r^n}{1-r} \)

Where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the total number of terms. This formula simplifies calculating large series sums, broadening your understanding of geometric progressions.

Understanding Mathematical Notation

Mathematical notation is a symbolic language used to represent numbers, functions, and operations in mathematics. It helps communicate complex ideas succinctly and is crucial when working with sequences and series.
In a geometric sequence problem, familiar symbols include:

• \( a \) for the first term
• \( r \) for the common ratio
• \( a_n \) for the \( n \)th term

Understanding these symbols allows you to follow the logic of formulas and expressions effortlessly. For instance, knowing that \( a_n = a \times r^{n-1} \) describes the \( n \)th term helps you see how each part of the equation contributes to finding any term in the sequence.
The use of exponents, fractions, and division in expressions like \( \frac{t^n}{2^{n-1}} \) demonstrates how mathematical notation can convey operations succinctly. By familiarizing yourself with these symbols, solving problems becomes more intuitive.