Problem 29 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the table feature of a graphing utility to find the first 10 terms of the sequences. (Assume \(n\) begins with 1.) $$a_{n}=1+\frac{n+1}{n}$$

Step-by-Step Solution

Answer

The first 10 terms of the sequence are \(a_{1} = 3\), \(a_{2} = 2.5\), \(a_{3} = 2.33\), \(a_{4} = 2.25\), \(a_{5} = 2.2\), \(a_{6} = 2.17\), \(a_{7} = 2.14\), \(a_{8} = 2.12\), \(a_{9} = 2.11\), \(a_{10} = 2.1\).

Step 1: Understand the sequence

The given sequence formula is \(a_{n}=1+\frac{n+1}{n}\). In this formula, replace \(n\) with each whole number starting from 1 up to 10 sequentially. This is how to generate the terms of the sequence.

Step 2: Compute each term of the sequence

Start with \(n=1\). Substitute 1 into the sequence formula to get the first term \(a_{1} = 1+\frac{1+1}{1} = 3\). Continue this process, plugging 2, 3, ..., 10 into the formula, and compute each corresponding term \(a_{n}\) respectively.

Step 3: Use the table feature of a graphing utility

In order to confirm the results and simplify the process, use the table feature of a graphing utility, such as a graphing calculator or an online graphing tool. Put \(y = 1+\frac{x+1}{x}\) in the function input field and get the corresponding y-values of x-values from 1 to 10, which are actually the first 10 terms of the sequence.

Key Concepts

Sequences

Sequences are an essential concept in mathematics, especially when dealing with series and patterns. A sequence is simply an ordered list of numbers where each number is called a term. Typically, these numbers follow a certain rule or pattern which is defined by a sequence formula. For example, the given sequence in our problem is defined by the formula \(a_{n} = 1 + \frac{n+1}{n}\). Here, each term is determined by substituting whole number values for \(n\). The sequence starts when \(n = 1\) and continues onwards, often indefinitely or until a specified endpoint. Each value of \(n\) corresponds to a specific term in the sequence, creating a predictable pattern based on the formula provided.

Sequence Formula

The sequence formula is crucial for generating the terms of a sequence. In our context, the given formula is \(a_{n} = 1 + \frac{n+1}{n}\). Let's break this down to understand it better:

• \(a_{n}\) - Represents the nth term of the sequence.
• \(1 + \frac{n+1}{n}\) - This expression calculates each term by adjusting \(n\). First, the numerator \(n+1\) is divided by \(n\), then this quotient is added to 1 to find \(a_{n}\).

Each term in the sequence is found by plugging different values of \(n\) into this formula. For instance, when \(n=1\), the calculation \(1 + \frac{2}{1}\) results in 3, which is the first term. This systematic calculation continues for each integer value of \(n\). The sequence formula is the engine of a sequence, precisely dictating the nature of its terms.

Graphing Calculator

A graphing calculator is a powerful tool that simplifies the generation and visualization of sequences. It allows you to input the sequence formula and provides a visual representation of each term's value. To use a graphing calculator for our sequence, you would:

• Enter the formula \(y = 1 + \frac{x+1}{x}\) into the calculator. Note that \(x\) is used instead of \(n\) because calculators typically use \(x\) for input variables.
• Access the table feature, which displays terms corresponding to different \(x\)-values. This means you can quickly generate values for a list of terms.

A graphing calculator thus streamlines the process. Instead of computing each term manually, the calculator retrieves them in seconds, verifying manual calculations and offering a broader perspective on the progression of the sequence.

Table Feature

The table feature in a graphing utility is immensely helpful for evaluating sequences, especially when checking work or exploring new patterns. It essentially creates a table of values from a function, in our case, the function \(y = 1 + \frac{x+1}{x}\):

• You start with a function that describes the sequence or pattern you are interested in.
• The table feature lists outputs (or terms) for an array of input values (like 1 to 10 for \(x\)).
• This saves a tremendous amount of time by automating the substitution process and presenting you with a clear framework of results.

Using the table feature, you can double-check the first 10 terms of a sequence, ensuring accuracy without manually going through potentially tedious calculations. It is a game changer for mathematics students and educators alike by adding efficiency and clarity to sequence analysis.