Problem 41 - College Algebra with Corequisite Support

Question

For the following exercises, write an explicit formula for each geometric sequence. \(a_{n}=\left\\{3,-1, \frac{1}{3},-\frac{1}{9}, \ldots\right\\}\)

Step-by-Step Solution

Answer

The explicit formula is \(a_n = 3 \cdot \left(-\frac{1}{3}\right)^{n-1}\).

Step 1: Identify the First Term

The first term of the geometric sequence is given. For this sequence, the first term \(a_1\) is 3.

Step 2: Find the Common Ratio

To find the common ratio \(r\), divide the second term by the first term: \(-1 \div 3 = -\frac{1}{3}\). Confirm by checking next terms: \(\frac{1}{3} \div (-1) = -\frac{1}{3}\). The common ratio \(r\) is confirmed as \(-\frac{1}{3}\).

Step 3: Write the Explicit Formula

In a geometric sequence, the nth term \(a_n\) can be written as \(a_n = a_1 \cdot r^{n-1}\). Using our known values ​​\(a_1 = 3\) and \(r = -\frac{1}{3}\), the explicit formula becomes \(a_n = 3 \cdot \left(-\frac{1}{3}\right)^{n-1}\).

Key Concepts

Explicit Formula

The explicit formula is a powerful tool in geometric sequences. It allows us to directly determine any term in the sequence without needing to know the previous ones. This formula is particularly useful when seeking distant terms in the sequence, as it skips repetitive calculations.

In a geometric sequence, the explicit formula can be given as \( a_n = a_1 \cdot r^{n-1} \). Here, \( a_n \) represents the term you want to find, \( a_1 \) is the first term of the sequence, \( r \) is the common ratio, and \( n \) is the term number. Thus, to find any term \( a_n \), you just need \( a_1 \) and \( r \).

For example, in the sequence given, the explicit formula \( a_n = 3 \cdot \left(-\frac{1}{3}\right)^{n-1} \) swiftly delivers any term meaning there is no need to count each step systematically.

Common Ratio

The common ratio in a geometric sequence is a constant factor between consecutive terms. This ratio is crucial because it dictates the direction and scale of the sequence. It can be found by dividing any term by its preceding term.

In our given sequence, the common ratio \( r \) is calculated as \( -1 \div 3 = -\frac{1}{3} \). This value is confirmed by checking the division between other consecutive pairs, such as \( \frac{1}{3} \div -1 \), which also gives \( -\frac{1}{3} \).

• The common ratio shows if the sequence is growing or shrinking.
• When the ratio is negative, the terms will alternate in sign.
• A ratio greater than 1 implies growth, while between 0 and 1 implies a decrease in magnitude.

First Term

The first term of a geometric sequence is the starting point from which the sequence unfolds. It is denoted as \( a_1 \), and every term in the sequence is built by multiplying this initial term by the common ratio raised to an increasing power.

In our sequence, the first term is given as \( a_1 = 3 \). This number initiates the sequence and helps determine all subsequent terms.

• The first term is essential for calculating terms using the explicit formula.
• It provides insight into the sequence's initial value and positioning.
• Coupled with the common ratio, it defines the sequence's complete behavior.

Remember, knowing \( a_1 \) and the common ratio allows for the generation of every term in the sequence through the explicit formula.