Precalculus (6th Edition) Blitzer

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

Chapter 10 - Section 10.3 - Geoetric Sequences and Series - Exercise Set - Page 1075: 83

Answer

$\$30000$.

Work Step by Step

For the periodic sum deposit, we calculate the annuity by: $A=\frac{P\left[ {{\left( 1+\frac{r}{n} \right)}^{nt}}-1 \right]}{\left( \frac{r}{n} \right)}$ From the given information, $P=\$1500$ and $r=0.05$ for $t=20\,\text{years}$. It is compounded annually, hence $n=1$. Thus: $\begin{align} & A=\frac{1500\left[ {{\left( 1+\frac{0.05}{1} \right)}^{20}}-1 \right]}{\left( \frac{0.05}{1} \right)} \\ & =\frac{1500\left[ {{\left( 1.05 \right)}^{20}}-1 \right]}{\left( 0.05 \right)} \\ & =\frac{1500\left[ \left( 2.653 \right)-1 \right]}{\left( 0.05 \right)} \\ & =49590 \end{align}$ For the lump sum deposit, we need to calculate the amount that is given by; $A=P\left[ {{\left( 1+\frac{r}{n} \right)}^{nt}} \right]$ For the lump sum deposit $P=\$30000$ and $r=5\%$ and it is compounded annually. Thus: $\begin{align} & A=30000\left[ {{\left( 1+\frac{0.05}{1} \right)}^{20}} \right] \\ & =30000\left[ 2.653 \right] \\ & =79590 \end{align}$ The difference between the lump sum and periodic deposit is: $amount-annuity=79590-49590=30000$
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