Problem 60 - Contemporary Precalculus

Question

Use a calculator to approximate the required term or sum. $$a_{50} \text { where } a_{n}=\frac{\ln n}{n^{2}}$$

Step-by-Step Solution

Answer

Answer: To find the approximate value of the 50th term, substitute n with 50 in the general term formula, which gives $$a_{50}=\frac{\ln 50}{50^{2}}$$. Using a calculator, we can find that the approximate value of the 50th term is around 0.00139.

Step 1: Identify the term's number in the sequence

We are given the general term: $$a_{n}=\frac{\ln n}{n^{2}}$$ We need to find the 50th term, so we will substitute the value of n with 50.

Step 2: Substitute the value of n into the formula

Now that we know we're looking for the 50th term, we can substitute n with 50 in the general term formula. $$ a_{50}=\frac{\ln 50}{50^{2}} $$

Step 3: Compute the value

All that is left now is to compute this value. Since we are allowed to use a calculator, we can easily find the approximation. $$ a_{50} \approx \frac{\ln 50}{50^{2}} $$ Enter the expression into the calculator and obtain the approximate value.

Step 4: Interpret the result

The calculator will provide the approximate value for the 50th term of the given sequence. This value represents the 50th term in a sequence where the general term is expressed as $$a_{n}=\frac{\ln n}{n^{2}}$$.

Key Concepts

General Term of a Sequence

In precalculus, understanding the general term of a sequence is fundamental to analyzing its properties and behavior. The general term, often denoted as a_n, represents a formula that can calculate any term in the sequence based on its position, n.

For example, with the general term a_n = \(\frac{\ln n}{n^{2}}\), you can find any term in the sequence just by plugging in the value of n, where n is a positive integer. So if you wanted the 50th term, identified as a₅₀, you'd substitute 50 for n in the formula to calculate its value. This approach is a methodical and efficient way to handle sequences without listing all of its terms.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions and have widespread applications in science, engineering, and mathematics. The natural logarithm, denoted as \(\ln(x)\), uses the base \(e\), where e is an irrational and transcendental number approximately equal to 2.71828.

Logarithms have unique properties that make them useful for solving equations involving exponential growth or decay. For instance, they can transform multiplicative relationships into additive ones, making certain classes of problems easier to tackle. In the context of sequences, logarithms can be used within the general term to create sequences that grow or decay at rates proportional to the reciprocal of the term's position, like in the example \(a_n = \frac{\ln n}{n^2}\).

Sequence Term Approximation

When dealing with sequences, especially those whose general term involves irrational numbers or complex operations, finding an exact value for specific terms may be impractical or impossible. Approximation becomes crucial in understanding the behavior of the sequence.

To approximate a term in a sequence like a₅₀ of \(a_n = \frac{\ln n}{n^2}\), one might use a scientific calculator or a software tool capable of computing logarithms. Approximation gives us a practical value to work with and can be particularly useful when the sequence describes a real-world scenario where an exact number is not necessary, or when contributing to a summation where the precise value may have a negligible effect on the total.