Chapter 5 - Number Theory and the Real Number System - 5.7 Arithmetic and Geometric Sequences - Exercise Set 5.7 - Page 331: 134

shown below

(a) For the population growth to be geometric, the ratio of adjacent term must be constant so, \[\begin{align} & r=\frac{21.27}{20.85} \\ & =1.02 \\ & =\frac{21.70}{21.27} \\ & =1.02 \end{align}\] And, \[\begin{align} & r=\frac{22.13}{21.70} \\ & =1.02 \\ & =\frac{22.57}{22.13} \\ & =1.02 \end{align}\] And, \[\begin{align} & r=\frac{23.02}{22.57} \\ & =1.02 \\ & =\frac{23.48}{23.02} \\ & =1.02 \end{align}\] \[\begin{align} & r=\frac{23.95}{23.48} \\ & =1.02 \\ & =\frac{24.43}{23.95} \\ & =1.02 \end{align}\] \[\begin{align} & r=\frac{24.92}{24.43} \\ & =1.02 \\ & =\frac{25.15}{24.92} \\ & =1.02 \end{align}\] We see that the common ratio is constant so population growth is geometric with common ratio \[1.02\] (b) As shown in part (a), population growth is geometric with,\[a=20.85,r=1.02\] By the formula of the \[{{n}^{th}}\] term of G.P. \[\begin{align} & {{a}_{n}}=a{{r}^{n-1}} \\ & {{a}_{n}}=20.85{{\left( 1.02 \right)}^{n-1}} \end{align}\] (c) The population in 2020 can be obtained by using \[n=21\] in the general expression of population. Because the population in 2020 will be \[{{21}^{st}}\] term of the geometric sequence modelling the population. \[\begin{align} & {{a}_{21}}=a{{r}^{20}} \\ & =20.85{{\left( 1.02 \right)}^{20}} \\ & {{a}_{21}}=30.98 \\ & =31 \end{align}\] In 2020, the population of Texas will be \[31\text{ million}\].