91Ó°ÊÓ

Insert three arithmetic means between 20 and 56

Population Growth: One of the most famous and controversial references to arithmetic and geometric progressions was made by Thomas Malthus in \(1798 .\) He wrote: "Population, when unchecked, increases in a geometrical ratio, and subsistence for man in an arithmetical ratio." Each day the size of a certain colony of bacteria is \(25 \%\) larger than on the preceding day. If the original size of the colony was 10,000 bacteria, find its size after 5 days.

Musical Scale: The frequency of the "A" note above middle C is, by international agreement, equal to \(440 \mathrm{Hz}\). A note one octave higher is at twice that frequency, or \(880 \mathrm{Hz}\). The octave is subdivided into 12 half-tone intervals, where cach half-tone is higher than the one preceding by a factor equal to the twelfth root of \(2 .\) This is called the equally tempered scale and is usually attributed to Johann Sebastian Bach \((1685-1750) .\) Write a GP showing the frequency of each half-tone, from 440 to 880 Hz. Work to two decimal places.

Show that the harmonic mean between two numbers \(a\) and \(b\) is given by $$\text { Harmonic Mean }=\frac{2 a b}{a+b}$$

Insert two harmonic means between \(\frac{7}{9}\) and \(\frac{7}{15}\)

Energy Consumption: If the U.S. energy consumption is \(7.00 \%\) higher each year, by what factor will the energy consumption have increased after 10.0 years?

Inflation: The price of a certain house, now \(\$ 126,000,\) is expected to increase by \(5 \%\) each year. Write a GP whose terms are the value of the house at the end of each year, and find the value of the house after 5 years.