Definition 2.4.2.1.
A geometric progression is a sequence of the form
\begin{equation*}
a, ar, ar^2, ar^3, \dots, ar^n, ar^{n + 1}, \dots ,
\end{equation*}
where the initial term \(a\) and common ratio \(r\) are real numbers.
Subsection 2.4.2 Geometric and arithmetic sequences
Suppose \(\seq{a_n}{n}\) is a geometric progression. How do we compute its initial term and common ratio? The initial term is the first term in the sequence. The common ratio is the ratio of consecutive terms:
\begin{equation*}
\frac{a_{n + 1}}{a_n} = \frac{ar^{n + 1}}{ar^n} = r,
\end{equation*}
assuming the sequence is not the zero sequence.
This is also how we can tell if a sequence is a geometric progression. A geometric progression must have the ratio of consecutive terms constant, no matter where we look in the sequence.
Example 2.4.2.2. Geometric progression.
The sequence \(3, 6, 12, 24, \dots\) is a geometric progression with initial term \(a = 3\text{.}\) The common ratio is \(r = 2\text{,}\) since
\begin{equation*}
\frac{6}{3} = \frac{12}{6} = \frac{24}{12} = \dots = 2.
\end{equation*}
Example 2.4.2.3. Geometric progression.
The sequence \(\seq{a_n}{n} = 2 \cdot 5^n\) for \(n = 0, 1, 2, \dots\) is a geometric progression with initial term \(a = 2\text{,}\) since \(a_0 = 2 \cdot 5^0 = 2\text{.}\) The common ratio \(r = 5\text{,}\) since
\begin{equation*}
\frac{a_{n+1}}{a_{n}} = \frac{2 \cdot 5^{n + 1}}{2 \cdot 5^n} = 5.
\end{equation*}
Example 2.4.2.4. Not a geometric progression.
The sequence \(1, 3, 5, 7, 9, \dots\) is not a geometric progression since the ratio of consecutive terms is not constant. For example,
\begin{equation*}
\frac{3}{1} \neq \frac{5}{3}.
\end{equation*}
Definition 2.4.2.5.
An arithmetic progression is a sequence of the form
\begin{equation*}
a, a + d, a + 2d, \dots, a + nd, \dots ,
\end{equation*}
where the initial term \(a\) and common difference \(d\) are real numbers.
Suppose \(\seq{a_n}{n}\) is an arithmetic progression. How do we compute its initial term and common difference? The initial term is the first term in the sequence. The common difference is the difference of consecutive terms:
\begin{equation*}
a_{n + 1} - a_n = (a + (n + 1)d) - (a + nd) = d.
\end{equation*}
This is also how we can tell if a sequence is an arithmetic progression. An arithmetic progression must have the difference of consecutive terms constant, no matter where we look in the sequence.
Example 2.4.2.6. Arithmetic progression.
The sequence \(2, 5, 8, 11, 14, \dots\) is an arithmetic progression with initial term \(2\text{.}\) The common difference is \(3\text{,}\) since
\begin{equation*}
5 - 2 = 8 - 5 = 11 - 8 = 14 - 11 = \dots = 3.
\end{equation*}
Example 2.4.2.7. Arithmetic progression.
The sequence \(\seq{a_n}{n}\) defined by \(a_n = 3 + 5n\) for \(n = 0, 1, 2, \dots\) is an arithmetic progression with initial term \(a = 3\text{,}\) since \(a_0 = 3 + 5 \cdot 0 = 3\text{.}\) The common difference is \(d = 5\text{,}\) since
\begin{equation*}
a_{n + 1} - a_n = (3 + 5(n + 1)) - (3 + 5n) = 5.
\end{equation*}
Example 2.4.2.8. Not an arithmetic progression.
The sequence \(1, 2, 4, 8, 16, 32, \dots\) is not an arithmetic progression, since the difference of consecutive terms is not constant. For example,
\begin{equation*}
2 - 1 \neq 4 - 2.
\end{equation*}
