Answer
a)
If the population growth is geometric then
\[\begin{align}
& r=\frac{34.21}{33.87} \\
& =1.01 \\
& =\frac{34.55}{34.21} \\
& =1.01
\end{align}\]
And,
\[\begin{align}
& r=\frac{34.90}{34.55} \\
& =1.01 \\
& =\frac{35.25}{34.90} \\
& =1.01
\end{align}\]
And,
\[\begin{align}
& r=\frac{35.60}{35.25} \\
& =1.01 \\
& =\frac{36}{35.60} \\
& =1.01
\end{align}\]
And,
\[\begin{align}
& r=\frac{36.36}{36} \\
& =1.01 \\
& =\frac{36.72}{36.36} \\
& =1.01
\end{align}\]
And,
\[\begin{align}
& r=\frac{37.09}{36.72} \\
& =1.01 \\
& =\frac{37.25}{37.09} \\
& =1.01
\end{align}\]
We see that the common ratio is constant so population growth is geometric with common ratio \[1.01\].
(b)
For the geometric growth of the population,\[a=33.87,r=1.01\]
By the formula of the \[{{n}^{th}}\] term of G.P.
\[\begin{align}
& {{a}_{n}}=a{{r}^{n-1}} \\
& {{a}_{n}}=33.87{{\left( 1.01 \right)}^{n-1}}
\end{align}\]
General term of the geometric sequence modelling the California’s population is\[{{a}_{n}}=33.87{{\left( 1.01 \right)}^{n-1}}\].
(c)
In 2020, the term of the G.P. will be \[{{21}^{st}}\] so,
\[\begin{align}
& {{a}_{21}}=a{{r}^{20}} \\
& =33.87{{\left( 1.01 \right)}^{20}} \\
& {{a}_{21}}=41.33
\end{align}\]
In 2020, the population of California will be \[41.33\]million.
