College Algebra (10th Edition)

Published by Pearson
ISBN 10: 0321979478
ISBN 13: 978-0-32197-947-6

Chapter 9 - Section 9.1 - Sequences - 9.1 Assess Your Understanding: 48

Answer

The first five terms are: $a_1= \sqrt2$ $a_2 = \dfrac{\sqrt{2\sqrt2}}{2}$ $a_3 = \dfrac{\sqrt[4]{2\sqrt2}}{2}$ $a_4 = \dfrac{\sqrt[8]{2\sqrt2}}{2}$ $a_5= \dfrac{\sqrt[16]{2\sqrt2}}{2}$

Work Step by Step

Use the given recursive formula to find the next four terms: $a_1= \sqrt2$ $a_2 = \sqrt{\dfrac{a_1}{2}} = \sqrt{\frac{\sqrt2}{2}}=\sqrt{\frac{\sqrt2}{2}\cdot \frac{2}{2}}=\dfrac{\sqrt{2\sqrt2}}{2}$ $a_3 = \sqrt{\dfrac{a_2}{2}}=\sqrt{\dfrac{\frac{\sqrt{2\sqrt2}}{2}}{2}}=\sqrt{\dfrac{\sqrt{2\sqrt2}}{4}}=\dfrac{\sqrt{\sqrt{2\sqrt2}}}{2}=\dfrac{\sqrt[4]{2\sqrt2}}{2}$ $a_4 = \sqrt{\dfrac{a_3}{2}}=\sqrt{\dfrac{\frac{\sqrt[4]{2\sqrt2}}{2}}{2}}=\sqrt{\dfrac{\sqrt[4]{2\sqrt2}}{4}}=\dfrac{\sqrt{\sqrt[4]{2\sqrt2}}}{2}=\dfrac{\sqrt[8]{2\sqrt2}}{2}$ $a_5= \sqrt{\dfrac{a_4}{2}}=\sqrt{\dfrac{\frac{\sqrt[8]{2\sqrt2}}{2}}{2}}=\sqrt{\dfrac{\sqrt[8]{2\sqrt2}}{4}}=\dfrac{\sqrt{\sqrt[8]{2\sqrt2}}}{2}=\dfrac{\sqrt[16]{2\sqrt2}}{2}$
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