Problem 62 - Contemporary Precalculus

Question

Use a calculator to approximate the required term or sum. $$a_{125} \text { where } a_{n}=\sqrt[n]{n}$$

Step-by-Step Solution

Answer

Answer: The 125th term of the sequence is approximately 1.04713.

Step 1: Find the term we need to calculate

We are given the formula for the sequence: $$a_n = \sqrt[n]{n}$$ We need to find the 125th term, which is $$a_{125} = \sqrt[125]{125}$$

Step 2: Calculate the 125th term using a calculator

Now, we need to use a calculator to approximate the value of $$a_{125}$$. You can use any scientific calculator or online calculator to do this.

Step 3: Interpret the result

After calculating the 125th term, you should get an approximate value of: $$a_{125} \approx 1.04713$$ So, the required term of the given sequence is approximately 1.04713.

Key Concepts

nth root

The concept of the \( n \)th root is a fundamental aspect of sequences and series, especially when dealing with expressions like \( \sqrt[n]{n} \). The \( n \)th root of a number \( x \) is essentially the number that, when raised to the power of \( n \), gives \( x \). This can be expressed mathematically as \( y^n = x \), where \( y \) is the \( n \)th root.

In the context of the exercise, the formula \( a_n = \sqrt[n]{n} \) requires finding the \( 125 \)th root of 125 to identify the 125th term, \( a_{125} \). Understanding nth roots is crucial because the result of applying this function gives us the patterns or terms in sequences like the one given.

To compute an \( n \)th root manually would involve finding a number that meets the power requirement exactly. However, since roots can be irrational or difficult to exact manually, we rely on calculators to provide precise approximations in these scenarios.

scientific calculator

A scientific calculator is a powerful tool that can handle various complex mathematical functions, including the computation of \( n \)th roots, exponents, and logarithms. These calculators are essential for finding precise approximations of terms in sequences or series that involve roots or powers.

When you need to approximate the 125th root of 125, a scientific calculator simplifies this task by allowing you to enter the expression \( \sqrt[125]{125} \) directly. Here's a common way to do it:

• Enter the number (125 in this case).
• Use the root function, often marked as \( \sqrt[x]{y} \) or accessed via a shift or function key.
• Input the root degree (125).
• Receive the calculated approximation.

Online calculators often work similarly, with intuitive interfaces to assist you in inputting the expressions correctly. This convenience helps in obtaining the result quickly and accurately, which in our exercise was approximately 1.04713.

approximations

Approximations in mathematics allow us to estimate the value of complex expressions or functions that are difficult or impossible to calculate precisely. In our exercise, the term \( a_{125} = \sqrt[125]{125} \) cannot be easily simplified or solved manually to an exact value.

This is where approximations come in handy. By using a scientific calculator, we reach a value that is very close to the true result, in this case, approximately 1.04713. Such an approximation is essential when dealing with real-world problems where exact answers may not be computationally feasible.

Approximations are also vital in fields such as engineering, physics, and computer science, where decisions need to be based on similarly calculated results. The precision of an approximation can depend on the technique used and the computational power available, but always strives to maintain a balance between efficiency and accuracy.