Problem 10 - Precalculus Mathematics for Calculus

Question

Find the \(n\) th term of the geometric sequence with given first term \(a\) and common ratio \(r .\) What is the fourth term? $$a=-6, \quad r=3$$

Step-by-Step Solution

Answer

The fourth term of the geometric sequence is -162.

Step 1: Understand the Geometric Sequence Formula

The formula for finding the nth term of a geometric sequence is \( a_n = a \cdot r^{n-1} \). Here, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the term number we want to find.

Step 2: Substitute Values into the Formula

We are given \( a = -6 \) and \( r = 3 \). We need to find the 4th term of the sequence, so \( n = 4 \). Substitute these values into the formula: \( a_4 = -6 \cdot 3^{4-1} \).

Step 3: Simplify the Exponent

Calculate the exponent: \( 3^{4-1} = 3^3 \). This simplifies to \( 3^3 = 27 \).

Step 4: Calculate the 4th Term

Substitute \( 27 \) back into the equation: \( a_4 = -6 \cdot 27 \). Now, multiply: \( -6 \cdot 27 = -162 \).

Step 5: Verify the Answer

Double-check that each calculation step used the correct values and operations to verify that the 4th term is correctly calculated.

Key Concepts

nth term formula

The nth term formula is a crucial concept in understanding geometric sequences. It provides a way to find a specific term in the sequence without having to list out all the previous terms. The formula is expressed as \[a_n = a \cdot r^{n-1}.\]This formula can be broken down into:

• \(a\): The first term of the sequence. It acts as the starting point from which the entire sequence builds.
• \(r^{n-1}\): This represents the multiplication of the common ratio \(r\) raised to the power of \(n-1\). The \(n-1\) exponent specifies how many times you have to multiply by the common ratio to reach the nth term from the first term.

Using this formula allows quick computation of any term in the sequence. In our given example, we calculated the 4th term by substituting \(a = -6\), \(r = 3\), and \(n = 4\) into the formula to find \(a_4\). This resulted in \(-6 \cdot 3^3\), ultimately leading to the value of \(-162\). It showcases the efficiency of using the nth term formula to directly calculate desired terms in a geometric sequence.

common ratio

The common ratio is a fundamental element of geometric sequences. It defines the relationship between consecutive terms in the sequence. In a geometric sequence, each term after the first is obtained by multiplying the preceding term by the common ratio \(r\). This consistent factor explains how values grow or shrink from one term to the next.

• The value of \(r\) determines the nature of the progression. A positive \(r\) results in each term being a positive multiple of the previous term, while a negative value affects the sign of the succeeding terms.
• If \(r\) is greater than 1, the sequence will grow exponentially. If it lies between 0 and 1, the sequence will decrease, becoming smaller and approaching zero. Negative \(r\) values introduce alternating signs within the sequence.

In our original exercise, the common ratio \(r\) was 3, showing that each term in the sequence is tripled from the preceding one. By substituting it into the nth term formula \(a_n = a \cdot r^{n-1}\), we saw how the progression rapidly increases within just four terms.

geometric progression

A geometric progression is an ordered set of numbers that follow a specific pattern defined by constant multiplication. It is a type of sequence where each term is derived by multiplying the preceding term by a fixed, non-zero value known as the common ratio.

• One characteristic of geometric progressions is that they can grow very quickly (exponentially). This happens when the common ratio, \(r\), is greater than 1.
• In contrast, if \(0 < r < 1\), the values of the progression decrease progressively towards zero, which can be called decay or shrinking effect.
• Geometric progressions are visually depicted as curves when plotted, due to their multiplier effect either increasing or decreasing at a consistent rate.

This type of progression is used in many fields, such as finance, physics, and computer science, to model scenarios where quantities grow or decay exponentially. In our specific problem, the sequence starting with \(a = -6\) and \(r = 3\), forms a geometric progression with terms growing rapidly through multiplication by three for each step.