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: 132


The salary for the sixth year will be\[\$38,288\].

Work Step by Step

In the first year, the salary of the player is $30,000 and annual year the increment is 5%. So, \[\begin{align} & 5%\text{ of }30000=\frac{5}{100}\times 30000 \\ & =5\times 300 \\ & =1500 \end{align}\] So, the salary of the player at the starting of the second year is: \[\$30,000+\$1,500=\$31,500\] At the starting of the third year, the salary of the player will be again increased by 5%. \[\begin{align} & 5%\text{ of }31500=\frac{4}{100}\times 31500 \\ & =4\times 315 \\ & =1260 \end{align}\] The salary for the third year is: \[\$31,500+\$1,260=\$32,760\] Now the series is 30000, 31500, 32760……. It is a form of G.P The nth term is found in G.P with the help of the following formula \[{{a}_{n}}=a{{r}^{\left( n-1 \right)}}\] The salary of the player accumulates according to G.P. with \[a=30000,\,r=1\cdot 05\] and it is required to find \[{{6}^{th}}\]term. \[\begin{align} & {{a}_{n}}=a{{r}^{n-1}} \\ & {{a}_{6}}=30000\times {{\left( 1.05 \right)}^{\left( 6-1 \right)}} \\ & =30000\times {{\left( 1.05 \right)}^{5}} \\ & {{a}_{6}}=38288.44 \end{align}\] Hence, the salary in year 6 rounded off to the nearest dollar will be\[\$38,288\].
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