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Chapter 13: Problem 10

Write the first five terms of each sequence. $$ a_{n}=\frac{n+2}{n} $$

Short Answer

Expert verified
The first five terms are 3, 2, \(\frac{5}{3}\), \(\frac{3}{2}\), and \(\frac{7}{5}\).

Step by step solution

01

- Identify the Sequence Formula

The given sequence formula is: \(a_{n}=\frac{n+2}{n}\)
02

- Calculate the First Term

Substitute \(n = 1\) into the formula: \(a_{1}=\frac{1+2}{1}=\frac{3}{1}=3\)
03

- Calculate the Second Term

Substitute \(n = 2\) into the formula: \(a_{2}=\frac{2+2}{2}=\frac{4}{2}=2\)
04

- Calculate the Third Term

Substitute \(n = 3\) into the formula: \(a_{3}=\frac{3+2}{3}=\frac{5}{3}\)
05

- Calculate the Fourth Term

Substitute \(n = 4\) into the formula: \(a_{4}=\frac{4+2}{4}=\frac{6}{4}=\frac{3}{2}\)
06

- Calculate the Fifth Term

Substitute \(n = 5\) into the formula: \(a_{5}=\frac{5+2}{5}=\frac{7}{5}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sequence Formulas
Understanding sequence formulas is crucial for solving problems involving algebraic sequences. A sequence formula provides a way to find any term in the sequence based on its position or index. In this exercise, the sequence formula is given as \(a_{n} = \frac{n+2}{n}\). This means the value of any term in the sequence can be found by plugging the term's position (\(n\)) into this equation. Familiarizing yourself with how to interpret and use these formulas can simplify the process of finding specific terms in any sequence.
Term Calculation
Calculating terms in a sequence is straightforward when you follow a systematic approach.
Let's break down the process using the formula from the exercise. To find any term \(a_{n}\), follow these steps:
  • Identify the position \(n\) of the term you want to find.
  • Substitute this value into the sequence formula.
  • Simplify the resulting expression to find the term.

For example, to calculate the first term \(a_{1}\), substitute \(n = 1\) into the formula: \(a_{1} = \frac{1+2}{1} = 3\). Repeating these steps for subsequent values of \(n\) will yield the respective terms in the sequence.
Rational Expressions
Many sequence formulas, like the one used in this exercise, result in rational expressions. A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding how to simplify these expressions is vital.

For instance, the given formula \(a_{n} = \frac{n+2}{n}\) is a rational expression. Simplification is often straightforward but important:
  • Divide the polynomials in the numerator and denominator whenever possible.
  • Reduce fractions to their simplest form.

In this sequence, when \(n = 4\), the term is \(a_{4} = \frac{4+2}{4} = \frac{6}{4} = \frac{3}{2}\). Simplifying rational expressions helps clarify each term and makes it easier to see patterns within the sequence.

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