Problem 44 - Precalculus with Limits

Question

In Exercises 35 - 44, write an expression for the \( n \)th term of the geometric sequence. Then find the indicated term. \( a_1 = 1000, r = 1.005, n = 60 \)

Step-by-Step Solution

Answer

The 60th term of the geometric sequence is the result of \( 1000 \cdot (1.005)^{59} \). Solve this for the numerical answer.

Step 1: Identify the values from the problem statement

First, ensure that you understand what each of the provided numbers represent in the context of a geometric sequence. In this case, \(a_1 = 1000\) is the first term of the sequence, \(r = 1.005\) is the common ratio between successive terms, and \( n = 60 \) is the term number for which we need to find the value.

Step 2: Apply the formula for the nth term of a geometric sequence

Apply the formula for the nth term of a geometric sequence \( a_n = a_1 \cdot r^{(n-1)} \). Substitute the known values \( a_1 = 1000\), \( r = 1.005 \) and \( n = 60 \) into the formula, resulting in \( a_{60} = 1000 \cdot (1.005)^{59} \).

Step 3: Calculate the value of the nth term

Evaluate the expression \( 1000 \cdot (1.005)^{59} \) to calculate the 60th term of the sequence.

Key Concepts

nth term of a sequence

In mathematics, a sequence refers to an ordered list of numbers where each number is referred to as a term. In a geometric sequence, each term is found by multiplying the previous term by a specific value known as the common ratio. The formula for determining the nth term of a geometric sequence is represented as \( a_n = a_1 \cdot r^{(n-1)} \). Here, \( a_n \) signifies the nth term you are seeking, \( a_1 \) is the first term of the sequence, \( r \) is the common ratio, and \( n \) is the position number of the sequence term you are trying to find.

Using this formula means you can find any term in the sequence by only knowing the first term, the common ratio, and the position number. It allows us to skip over others and calculate directly the term we are interested in, which is useful especially when working with a long sequence.

common ratio

The common ratio is vital in defining a geometric sequence. It is the factor by which each term in the sequence is multiplied to get the next term. In the formula for the nth term, it is denoted as \( r \).

For example, in the sequence given where the first term \( a_1 \) is 1000 and the common ratio \( r \) is 1.005, each term is 1.005 times the previous one.

The common ratio helps determine how quickly a sequence grows or shrinks:

• If \( r > 1 \), the terms increase.
• If \( 0 < r < 1 \), the terms decrease.
• If \( r = 1 \), the sequence is constant.
• If \( r < 0 \), the terms alternate in sign.

Understanding the common ratio is essential to predicting and analyzing the behavior of a geometric sequence.

geometric progression

A geometric progression, or geometric sequence, is a series of numbers where each term after the first is the product of the previous one and a fixed number known as the common ratio. It's a progression where multiplication, rather than addition or subtraction, dictates the relationship between terms.

In our example, the sequence starts at 1000, and each subsequent term grows by the factor of 1.005. This could represent, for instance, a financial scenario where an investment grows at a steady rate of 0.5% per period. In this case, the geometric sequence allows us to model and project future values conveniently.

Geometric progressions are prevalent in various fields such as physics, biology, computer science, finance, and more. They help in modeling exponential growth or decay, like population growth, radioactive decay, or compound interest. Understanding this concept can help in visualizing changes that aren't linear, making it a powerful tool in analysis and prediction.