Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 12 - Sequences and Series - Chapter Review - Review Exercises - Page 662: 14

Answer

$${a_4} = 16,\,\,\,\,\,{a_n} = 128{\left( {\frac{1}{2}} \right)^{n - 1}}{\text{ and }}{S_5} = 248$$

Work Step by Step

$$\eqalign{ & {a_1} = 128,\,\,\,\,r = 1/2 \cr & {\text{The general term of a geometric sequence is }}{a_n} = {a_1}{r^{n - 1}}.{\text{ }} \cr & {\text{substituting }}{a_1} = 128,\,\,\,\,r = 1/2 \cr & {a_n} = 128{\left( {\frac{1}{2}} \right)^{n - 1}} \cr & \cr & {\text{find }}{a_4},{\text{ substitute }}n = 4{\text{ into the general term formula}} \cr & {a_4} = 128{\left( {\frac{1}{2}} \right)^{4 - 1}} \cr & {a_4} = 128\left( {\frac{1}{8}} \right) \cr & {a_4} = 16 \cr & \cr & {\text{then the sum of the first }}n{\text{ terms}}{\text{, is given by}} \cr & {S_n} = \frac{{{a_1}\left( {{r^n} - 1} \right)}}{{r - 1}},{\text{ where }}r \ne 1 \cr & {\text{For the first five terms we have}} \cr & {S_5} = \frac{{\left( {128} \right)\left( {{{\left( {1/2} \right)}^5} - 1} \right)}}{{1/2 - 1}} \cr & {S_5} = \frac{{\left( {128} \right)\left( {1/32 - 1} \right)}}{{ - 1/2}} \cr & {S_5} = 248 \cr & \cr & {\text{then}} \cr & {a_4} = 16,\,\,\,\,\,{a_n} = 128{\left( {\frac{1}{2}} \right)^{n - 1}}{\text{ and }}{S_5} = 248 \cr} $$
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