## Thinking Mathematically (6th Edition)

(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 $31\text{ million}$.