Identify a geometric series

Bars that shrink by the same fraction

The first four bars have the heights 12, 9, 6.75 and 5.0625.

Divide each height by the height before it: \(\frac{9}{12}=0.75\), \(\frac{6.75}{9}=0.75\) and \(\frac{5.0625}{6.75}=0.75\).

The quotient is the same for every pair of neighbors, so each bar is 0.75 times the bar before it. Move the slider to join other pairs with the orange arrow.

A list of numbers with this property is a geometric sequence, and the sum of its terms is a geometric series.

The same ratio at every step

The first bar has the height \(a\), which is not 0, and each bar is \(r\) times the bar before it.

The heights are \(a\), \(ar\), \(ar^2\), \(ar^3\) and so on, and the number \(r\) is the common ratio.

Move the slider to change \(r\). If \(r\) is greater than 1, the bars grow. If \(r\) is between 0 and 1, the bars shrink.

If \(r\) is negative, each bar lies on the other side of the axis from the bar before it. The red bars lie below the axis, so they count as negative.

The bars change, but each bar divided by the bar before it is still \(r\), and this makes the series geometric.

Notation

1. A geometric series has this form. The first term is \(a\), and each term is \(r\) times the term before it.

   \[\cy{a}+\cy{a}\cq{r}+\cy{a}\cq{r}^2+\cdots=\sum_{n=0}^{\infty}\cy{a}\cq{r}^n\]

2. To find \(r\), divide any term \(a_{n+1}\) by the term \(a_n\) before it. The quotient must be the same for every pair of neighbors.

   \[\cq{r}=\dfrac{a_{n+1}}{a_n}\]

3. The number \(a\) is the first term that is in the sum. If the sum starts at a later index, \(a\) is the value of the term at that index.

4. If the formula for the \(n\)th term is not a number times \(r^n\), rewrite it, or find the first few terms and divide them.

When it fails

Each bar is now 3 less than the bar before it, so the heights are 12, 9, 6 and 3. The red arrows show the quotients 0.75, about 0.67, and 0.5.

The quotients are not equal, so this is not a geometric series. The first quotient is still 0.75, so a test on the first two terms alone does not find the difference.

Example 1: A negative ratio

Show that \(20-10+5-\frac52+\cdots\) is a geometric series. Find \(a\) and \(r\).

1. Divide each term by the term before it.

   \[\begin{aligned}\frac{-10}{20}&=-\frac12\\ \frac{5}{-10}&=-\frac12\\ \frac{-5/2}{5}&=-\frac12\end{aligned}\]

2. The three quotients are equal, so the series is geometric. The ratio is negative, so every second term is negative, and its bar is red, below the axis.

3. The first term is the height of the first bar, and the ratio is the common quotient.

   \[a=20,\qquad r=-\frac12\]

Example 2: A sum that starts at a later index

Find \(a\) and \(r\) for \(\displaystyle\sum_{n=3}^{\infty}5\left(\frac23\right)^n\).

1. The sum starts at \(n=3\), so the first three terms in it come from \(n=3\), 4 and 5.

   \[\begin{aligned}5\left(\tfrac23\right)^3&=\tfrac{40}{27}\\ 5\left(\tfrac23\right)^4&=\tfrac{80}{81}\\ 5\left(\tfrac23\right)^5&=\tfrac{160}{243}\end{aligned}\]

2. A tempting wrong move: The formula has the factor 5 in front, so the first term is \(a=5\).

   \[a=5\]

   The number 5 is the term at \(n=0\), the first outlined bar, and the outlined terms are not in the sum. The first bar in the sum has the height \(\frac{40}{27}\), about 1.48.

3. The first term is the term at the starting index. Divide the second term by the first term to find the ratio.

   \[\begin{aligned}a&=5\left(\tfrac23\right)^3=\tfrac{40}{27}\\ r&=\frac{80/81}{40/27}=\frac23\end{aligned}\]

Practice

Show that \(\displaystyle\sum_{n=1}^{\infty}\frac{3^{n+1}}{5^n}\) is geometric. Find \(a\) and \(r\).

Move the slider to show the first three terms as bars. Divide each term by the term before it. The first of the three is \(a\).

Questions

Questions 1 and 2 use the figure, which shows four bars with the heights 3, 6, 12 and 24.

1. What is the common ratio?

   • 2
   • 3
   • \(\dfrac12\)
   Answer

   2

2. If the pattern continues, what is the height of the fifth bar?

   • 48
   • 30
   • 36
   Answer

   48

3. The series \(\displaystyle\sum_{n=2}^{\infty}4\left(-\frac12\right)^n\) is geometric. What is its first term?

   • 1
   • 4
   • −2
   Answer

   1

Summary

You identify a geometric series by dividing each term by the term before it and checking that every quotient is the same.

A constant ratio means that each term is the previous term times one number, so the terms are \(a\), \(ar\), \(ar^2\) and so on.

You read the first term from the sum and the ratio from the quotient, before you test convergence or find a sum.

This is one of four skills in topic 10.2 of Which & Why: AP Calculus.