Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 10 - Review Exercises - Page 1125: 59


See the explanation below.

Work Step by Step

$\begin{align} & {{S}_{1}}:2=2{{\left( 1 \right)}^{2}} \\ & 2=2\ \text{is true}\text{.} \\ & {{S}_{k}}:2+6+10+....+\left( 4k-2 \right)=2{{k}^{2}} \\ & {{S}_{k+1}}:2+6+10+....+\left( 4k-2 \right)+\left( 4k+2 \right)=2{{\left( k+1 \right)}^{2}} \\ \end{align}$ And add $\left( 4k+2 \right)$ to both sides of ${{S}_{k}}$: $2+6+10+....+\left( 4k-2 \right)+\left( 4k+2 \right)=2{{k}^{2}}+\left( 4k+2 \right)$ Then, simplify the right-hand side: $\begin{align} & 2{{k}^{2}}+4k+2=2({{k}^{2}}+2k+1) \\ & 2{{k}^{2}}+4k+2=2{{\left( k+1 \right)}^{2}} \\ & \text{If }{{S}_{k}}\ \text{is true, then }{{S}_{k+1}}\ \text{is true}\text{. } \\ \end{align}$
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