Geometric Sequences
A Geometric Sequence is a list of values that is derived by multiplying the previous term by a constant, called the common ratio. We utilize \(r\) to represent the common ratio.
Example 1: Determine if the following sequence is a geometric sequences. If it is a geometric sequence, find the common ratio.
\(10,\ 5,\ \large\frac{5}{2},\ \frac{5}{4},\ \frac{5}{8},\ ...\)
In order to determine if the sequence is geometric, find the common ratio between consecutive terms: \(\large{r=\frac{a_n}{a_{n-1}}}\).
\(\large{r=\frac{5}{10}=\frac{1}{2}}\) and \(\large{r=\frac{\frac{5}{2}}{5}=\frac{1}{2}}\) and \(\large{r=\frac{\frac{5}{4}}{\frac{5}{2}}=\frac{1}{2}}\) and \(\large{r=\frac{\frac{5}{8}}{\frac{5}{4}}=\frac{1}{2}}\)
Since the ratio of consecutive terms is constant, the sequence is a geometric sequence.
The points \(\left(n,\ a_n\right)\) of the geometric sequence are shown in the graph below. Click on each point to see the coordinates. What type of curve do the coordinates of a geometric sequence lie on? If you cannot identify the curve, scroll down and click on the function as a hint.
A Geometric Sequence is a list of values that is derived by multiplying the previous term by a constant, called the common ratio. We utilize \(r\) to represent the common ratio.
Example 1: Determine if the following sequence is a geometric sequences. If it is a geometric sequence, find the common ratio.
\(10,\ 5,\ \large\frac{5}{2},\ \frac{5}{4},\ \frac{5}{8},\ ...\)
In order to determine if the sequence is geometric, find the common ratio between consecutive terms: \(\large{r=\frac{a_n}{a_{n-1}}}\).
\(\large{r=\frac{5}{10}=\frac{1}{2}}\) and \(\large{r=\frac{\frac{5}{2}}{5}=\frac{1}{2}}\) and \(\large{r=\frac{\frac{5}{4}}{\frac{5}{2}}=\frac{1}{2}}\) and \(\large{r=\frac{\frac{5}{8}}{\frac{5}{4}}=\frac{1}{2}}\)
Since the ratio of consecutive terms is constant, the sequence is a geometric sequence.
The points \(\left(n,\ a_n\right)\) of the geometric sequence are shown in the graph below. Click on each point to see the coordinates. What type of curve do the coordinates of a geometric sequence lie on? If you cannot identify the curve, scroll down and click on the function as a hint.
When the points \(\left(n,\ a_n\right)\) of a geometric sequence are plotted, they lie on an exponential curve. Remember, we are just plotting the points—not connecting the dots—because the n values cannot be negative, zero or fractions. Since \(n\) represents the position the value is in the sequence, it wouldn’t make sense to have a \(1.5th\) term or a \(-4th\) term.
Explicit Rule for a Geometric Sequence: \(a_n=a_1\left(r\right)^{n-1}\) where \(a_1\) represents the value of the first term and \(r\) is the common ratio.
Example 2: Write the explicit rule for the \(nth\) term of the geometric sequence \(3,\ 12,\ 48,\ 192,\ ...\) Then find the value of the \(10th\) term.
The common ratio is \(4\) \(\left(\frac{12}{3}=4\ ,\ \frac{48}{12}=4,\ \frac{192}{48}=4\right)\) and the first term is \(3\).
Since \(a_1=3\) and \(r = 4\), the explicit rule is \(a_n=3\left(4\right)^{n-1}\).
To find the \(10th\) term, substitute 10 in for \(n\).
\(a_{10}=3\left(4\right)^{10-1}\)
\(a_{10}=3\left(4\right)^{9}\)
\(a_{10}=786,432\)
Example 3: Find the explicit rule for the geometric sequence with \(r = 2\) and \(a_3=5\).
Example 4: Find the explicit rule for the geometric sequence with \(a_3=75\) and \(a_6=-9375\).