Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 5 - Number Theory and the Real Number System - Chapter Summary, Review, and Test - Review Exercises - Page 337: 146

Answer

The given sequence is an arithmetic sequence and the next two terms of the given sequence are\[-28\ \text{and}\ -35\].

Work Step by Step

To check is a sequence is an arithmetic sequence, see if the differences between two consecutive terms are equal i.e. check if\[{{a}_{n+1}}-{{a}_{n}}=d\ \ \] for all n element of N, here \[d\] is the common difference. To check if a sequence is a geometric sequence, see if the ratio between two consecutive terms is equal i.e. check if \[\frac{{{a}_{n+1}}}{{{a}_{n}}}=r\ \]for all n element of N here \[r\] is the common ratio. When\[n=1\], for the given sequence, \[\begin{align} & d={{a}_{2}}-{{a}_{1}} \\ & =-7-0 \\ & =-7 \end{align}\] And, \[\begin{align} & r=\frac{{{a}_{2}}}{{{a}_{1}}} \\ & =\frac{-7}{0} \\ & =\text{Not defined} \end{align}\] When\[n=2\], for the given sequence, \[\begin{align} & d={{a}_{3}}-{{a}_{2}} \\ & =-14-\left( -7 \right) \\ & =-7 \end{align}\] And, \[\begin{align} & r=\frac{{{a}_{3}}}{{{a}_{2}}} \\ & =\frac{-14}{-7} \\ & =2 \end{align}\] When\[n=3\], for the given sequence, \[\begin{align} & d={{a}_{4}}-{{a}_{3}} \\ & =-21-\left( -14 \right) \\ & =-7 \end{align}\] And, \[\begin{align} & r=\frac{{{a}_{4}}}{{{a}_{3}}} \\ & =\frac{-21}{-14} \\ & =1.5 \end{align}\] Since, \[d=-7\ \ \forall n=1,2,3\]and \[r\]is not equal for all n element of N it implies the given sequence is an arithmetic sequence. Use the formula \[{{a}_{n}}={{a}_{n-1}}+d\]to find the next term of the arithmetic sequence. Put \[n=5\]in\[{{a}_{n}}={{a}_{n-1}}+d\]to find the fifth term of the given arithmetic sequence as follows: \[\begin{align} & {{a}_{5}}={{a}_{5-1}}+d \\ & ={{a}_{4}}+d \\ & =-21-7 \\ & =-28 \end{align}\] Put \[n=6\]in\[{{a}_{n}}={{a}_{n-1}}+d\]to find the sixth term of the given arithmetic sequence as follows: \[\begin{align} & {{a}_{6}}={{a}_{6-1}}+d \\ & ={{a}_{5}}+d \\ & =-28-7 \\ & =-35 \end{align}\] The given sequence is an arithmetic sequence and the next two terms of the given sequence are\[-28\ \text{and}\ -35\].
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