# Chapter 7 - Section 7.2 - Arithmetic Sequences and Series - 7.2 Exercises - Page 645: 12

$3-\sqrt{2},3,3+\sqrt{2}, 3+2\sqrt{2},3+3\sqrt{2}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ To find the first $5$ terms, solve first the common difference. Then keep adding the common difference to the given second term. $\bf{\text{Solution Details:}}$ The common difference is the difference between a term and the term before it. Hence, the common difference, $d,$ is \begin{array}{l}\require{cancel} d=a_2-a_1 \\\\ d=3-(3-\sqrt{2}) \\\\ d=3-3+\sqrt{2} \\\\ d=\sqrt{2} ,\end{array} Adding the common difference, the third term is \begin{array}{l}\require{cancel} a_3=a_2+d \\\\ a_3=3+\sqrt{2} ,\end{array} the fourth term is \begin{array}{l}\require{cancel} a_4=a_3+d \\\\ a_4=3+\sqrt{2}+\sqrt{2} \\\\ a_4=3+2\sqrt{2} ,\end{array} and the fifth term is \begin{array}{l}\require{cancel} a_5=a_4+d \\\\ a_5=3+2\sqrt{2}+\sqrt{2} \\\\ a_5=3+3\sqrt{2} ,\end{array} Hence, the first $5$ terms are $3-\sqrt{2},3,3+\sqrt{2}, 3+2\sqrt{2},3+3\sqrt{2} .$

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