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, a1, and common ratio, r. Find \(a_{40}\) when \(a_{1}=1000, r=-\frac{1}{2}.\)

Step-by-Step Solution

Answer

The \(a_{40}\) given \(a_{1}=1000\) and \(r=-\frac{1}{2}\) is a small decimal number, very close to zero, and negative.

Step 1: Identify terms

Identify the first term, the common ratio and the term number. Here, the first term \(a_1\) is 1000, the common ratio \(r\) is -1/2, and we are looking for the 40th term, so \(n = 40\).

Step 2: Substitute values in Formula

Substitute these values into the formula for the nth term: \(a_n = a_1 * r^{n-1}\) to get \(a_{40} = 1000 * (-\frac{1}{2})^{40-1}\).

Step 3: Solve the equation

Next, calculate the power of -1/2: \( a_{40} = 1000 * (-\frac{1}{2})^{39}\). As the base is negative and the power is odd, the result would also be negative. Since 2 to the power 39 is a very large number, for simplicity, we can write it as: \( a_{40} = \frac{1000}{2^{39}} \times -1\).

Step 4: Simplify the expression

Simplify the expression to get the final answer. This would involve simple mathematical calculations and simplifications. The large denominator of \(2^{39}\) can be computed using a calculator to get the decimal value.

Key Concepts

General Term of a Geometric Sequence

A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

The general term of a geometric sequence can be written as \( a_n = a_1 \times r^{(n-1)} \), where \( a_n \) represents the nth term of the sequence, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number. This formula helps us find any term in the sequence without having to list all the preceding terms, which is particularly useful for finding terms later in the sequence, like the 40th term in our exercise example.

The general term formula embodies exponential growth or decay depending on whether the common ratio is greater than one or between zero and one respectively. It simplifies the process of sequence term calculation and illustrates the powerful nature of exponential expressions in sequences.

Common Ratio

The common ratio in a geometric sequence is the constant factor between consecutive terms. It is what characterizes the sequence as geometric. In the provided exercise, the common ratio is \( r = -\frac{1}{2} \), indicating that each term is half the previous term and also alternates signs because it is negative.

This value is critical as it dictates the behavior of the sequence, allowing the terms to either increase or decrease, and can also produce a sequence with alternating positive and negative terms when the common ratio is negative. Understanding the impact of the common ratio on the sequence is essential for comprehensive analysis and determination of the sequence's properties.

Exponential Expressions

Exponential expressions like \( r^{n-1} \) in the formula of the general term of a geometric sequence, demonstrate how quickly numbers can grow or decay.

The base of the exponential expression is the common ratio, and the exponent is one less than the term's position number. This is because the first term is not multiplied by the common ratio. Exponential expressions are not always straightforward to compute, especially with larger exponents or negative bases, as seen in the exercise. With a negative common ratio raised to an odd exponent, the result is negative, which affects the sign of the term found. This exercise highlights the importance of grasping exponential expressions to understand their implications in sequences and other mathematical contexts.

Sequence Term Calculation

To calculate a term in a geometric sequence, like \( a_{40} \) in our example, substitute the known values into the general term formula, evaluate the exponential expression, and then multiply by the first term. Here is a brief illustration:

Calculating the 40th term

1. Substitute the known values: \( a_{40} = 1000 \times (-\frac{1}{2})^{40 - 1} \) 2. Evaluate the exponential expression: \( (-\frac{1}{2})^{39} \) 3. Perform the multiplication to find the 40th term.
This simple process emphasizes the formula's practicality in finding specific terms quickly. For large values of \( n \), or for complex common ratios, this direct approach is much more efficient than constructing the entire sequence term by term.