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 - 5.7 Arithmetic and Geometric Sequences - Exercise Set 5.7 - Page 331: 147

Answer

The provided statement is false. If the arithmetic sequence is\[1,\ 4,\ 7,\ 10,\ 13,\ \ldots \], common difference is \[3\].

Work Step by Step

The sequence \[1,\ 4,\ 8,\ 13,\ 19,\ 26,\ \ldots \] is an arithmetic sequence. In the provided sequence,\[{{a}_{1}}=1,\ {{a}_{2}}=4,\ {{a}_{3}}=8,\ \ldots \] The common difference is \[\begin{align} & d={{a}_{2}}-{{a}_{1}} \\ & ={{a}_{3}}-{{a}_{2}} \\ & ={{a}_{4}}-{{a}_{3}} \end{align}\] and \[\begin{align} & {{d}_{1}}={{a}_{2}}-{{a}_{1}} \\ & {{d}_{1}}=4-1 \\ & =3 \end{align}\] and \[\begin{align} & {{d}_{2}}={{a}_{3}}-{{a}_{2}} \\ & =8-4 \\ & =4 \end{align}\] Hence, \[{{d}_{1}}\ne {{d}_{2}}\](common difference is not same). Hence, the provided sequence is not A.P. If the arithmetic sequence is\[1,4,7,10,13,\ldots \], common difference is \[3\]. Hence, the provided statement is false.
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