Problem 13 - College Algebra

Question

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, \(a_{1},\) and common ratio, \(r .\) Find \(a_{40}\) when \(a_{1}=1000, r=-\frac{1}{2}\)

Step-by-Step Solution

Answer

The 40th term of the sequence, \(a_{40} = 1000 \cdot -2^{-39}\). This involves a large exponent of a fraction, so it may be more palatable to represent as a decimal or in scientific notation as needed.

Step 1: Define the Given Variables

We have \(a_{1}=1000\) as the first term of the sequence, \(r=-\frac{1}{2}\) as the common ratio, and we're looking for the 40th term, or \(a_{40}\).

Step 2: Apply the n-th Term Formula for a Geometric Series

We now apply the formula for the n-th Term for a Geometric Series: \(a_{n}=a_{1}\cdot r^{n-1}\). We substitute the values into this formula: \(a_{40}=a_{1}\cdot r^{40-1}\) to find the 40th term.

Step 3: Calculate the 40th term of the Sequence

The equation transforms as follows:\n \(a_{40}=1000\cdot \left(-\frac{1}{2}\right)^{40-1}\)\n \(a_{40}=1000\cdot \left(-\frac{1}{2}\right)^{39}\)\n Simplify \(-\frac{1}{2}^{39}\)\n \(a_{40}=1000\cdot \left(\frac{-1}{2}\right)^{39}\)\n \(a_{40}=1000\cdot -2^{-39}\)

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This type of series is characterized by the fact that the ratio between successive terms remains constant.

In other words, each term is the product of the previous term and the common ratio, making it relatively straightforward to calculate any term in the series if you know the first term and the common ratio. Geometric series have significant applications in fields such as finance for calculating compound interest, in computer science for analyzing algorithms, and in physics for summing forces or other quantities that have a common ratio.

Common Ratio

The common ratio in a geometric sequence is the consistent factor between consecutive terms. This ratio is incredibly significant because it defines the behavior of the series. If the common ratio is greater than one, the terms of the series increase exponentially. If the ratio is a fraction, the terms decrease in size but never reach zero, illustrating a decay. And if the common ratio is negative, as it is in the exercise at hand, the terms alternate in sign, resulting in a sequence that oscillates between positive and negative values.

Understanding the common ratio provides insight into the growth or decay of a sequence and can also be helpful when deriving formulas for sums of a finite or infinite geometric series.

Sequence Term Calculation

The calculation of any term in a geometric sequence, known as the n-th term calculation, is pivotal for understanding how the sequence develops over time. The n-th term is given by the formula \(a_n = a_1 \cdot r^{n-1}\), where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term position in the sequence. This formula allows you to find any term without having to calculate all preceding terms, which can be highly efficient for large values of \(n\).

For instance, to find the 40th term in the given exercise, we apply the formula with the given values, resulting in \(a_{40} = 1000 \cdot (-\frac{1}{2})^{39}\). This demonstrates the power of exponential growth or decay inherent in geometric sequences and emphasizes the importance of understanding how to manipulate the n-th term formula correctly.