Problem 55 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the table feature of a graphing utility to find the first 10 terms of the sequence. (Assume \(n\) begins with 1.) $$a_{n}=1.5+0.05 n$$

Step-by-Step Solution

Answer

The first 10 terms of the sequence are: 1.55, 1.6, 1.65, 1.7, 1.75, 1.8, 1.85, 1.9, 1.95, 2

Step 1: Understand the Function

The function given describes a sequence where each term \(a_{n}\) can be found by plugging the term number \(n\) into the function \(a_{n}=1.5+0.05 n\). This sequence has a starting term of \(a_{1} = 1.5 + 0.05 \cdot 1\). Each subsequent term is 0.05 greater than the previous term.

Step 2: Compute the first 10 terms

To find the first 10 terms, plug each value from \(n = 1\) to \(n = 10\) into the function. The resulting sequence would therefore be: \(a_{1} = 1.5 + 0.05 \cdot 1\), \(a_{2} = 1.5 + 0.05 \cdot 2\), \(a_{3} = 1.5 + 0.05 \cdot 3\), ... , \(a_{10} = 1.5 + 0.05 \cdot 10\).

Step 3: Calculate the values

By calculating the terms based on the function, we find the first 10 terms of the sequence to be: 1.55, 1.6, 1.65, 1.7, 1.75, 1.8, 1.85, 1.9, 1.95, 2

Key Concepts

Graphing Utility

Graphing utilities are powerful tools that can simplify visualizing mathematical concepts, especially sequences and their properties. A graphing utility usually consists of a graphing feature, where users can plot functions and data points, and a table feature, where users can view numerical values of sequences or function outputs for different inputs.

In the context of arithmetic sequences, as in the exercise involving the function \(a_n = 1.5 + 0.05n\), a graphing utility can be used to quickly generate the first ten terms of the sequence without manual calculation. By inputting the function into the table feature of the graphing utility, it automatically computes the terms for successive values of \(n\), starting with 1. This enables students to effectively verify the terms they have derived algebraically or to find a pattern in the sequence which might not be immediately evident from the formula alone.

Function Notation

Understanding function notation is crucial for working with sequences and mathematical expressions in general. Function notation, involving symbols like \(f(x)\) or \(a_n\), describes the relationship between input values and their corresponding outputs. In the context of sequences, function notation specifically conveys how each term of the sequence is related to its position, often denoted by \(n\), within the sequence.

In the given exercise, the function is written as \(a_n = 1.5 + 0.05n\). This reveals that the sequence term \(a_n\) depends linearly on the term number \(n\). Function notation efficiently communicates the rule to find any term in the sequence without listing all terms outright. It also helps to understand the structure and behavior of the sequence as the term number increases.

Sequence Terms Calculation

The calculation of sequence terms is a fundamental aspect of understanding sequences in mathematics. An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant difference to the preceding term. The formula provided within the exercise, \(a_n = 1.5 + 0.05n\), represents an arithmetic sequence where 1.5 is the first term, and 0.05 is the common difference added to each subsequent term.

Calculating the terms of a sequence involves substituting successive integers into the function notation for \(n\). Hence, to get the first ten terms of this sequence, one would calculate \(a_1\), \(a_2\), ..., \(a_{10}\) by plugging values from 1 to 10 into the equation. Understanding how to perform these calculations is vital as it allows students to derive any term they need, providing a greater comprehension of sequences, and preparing them for further studies in series and mathematical patterns.