Geometric sequence
- Each term is the one before multiplied by the same number, the common ratio \(r\).
- \(u_n = u_1 r^{n-1}\)
- \(r\) can be negative (the signs alternate) or between 0 and 1 (the terms shrink).
12.1.3
Geometric Sequences
12.1.3
Geometric Sequences
By the end of this section
- Identify geometric sequences from their constant ratio
- Use the formula $u_n = u_1 \times r^{n-1}$ to find any term
- Find the first term and common ratio given two terms
- Work with negative and fractional ratios
- Understand the definition of geometric sequences
- Use the formula $u_n = u_1 r^{n-1}$ to find any term
- Determine the common ratio from given terms
- Apply geometric sequences to growth and decay problems
Key points
Percentage change
- A rise of \(p\%\) each period multiplies by \(1 + \frac{p}{100}\). A fall of \(p\%\) multiplies by \(1 - \frac{p}{100}\).
- After \(k\) periods the starting value has been multiplied \(k\) times.
On the calculator
A table of terms
Menu > Table, or a Lists & Spreadsheet page
- Enter the rule for \(u_n\) and scroll to the value you need.
Screenshot and video to come
Explore: arithmetic and geometric sequences
Choose arithmetic or geometric, then change the first term and d or r. Before you tick the box, predict what the sums Sₙ will look like.
| n | uₙ | Sₙ |
|---|
Move a slider to change the sequence.
- Arithmetic: what shape do the dots make, and what does d do to it? What happens when d is negative?
- Geometric: set r between 0 and 1, then below 0, then above 1. Describe the dots each time.
- Tick the sums. For which settings do the sums Sₙ climb in a straight line, and for which do they level off?
- Set r = 1 in the geometric sequence. What is Sₙ, and why can the formula not be used?
Worked examples
Problems like the ones in class. Pick one, try it, then open the working.Find a term
A geometric sequence has first term 5 and common ratio −3. Find the 8th term.
Working
Have a go first. Then open the working one line at a time.
-
The rule
\(u_n = u_1 r^{n-1} = 5(-3)^{n-1}\)
First term 5, ratio −3.
-
The 8th term
\(u_8 = 5(-3)^7 = -10\,935\)
Seven multiplications by −3. An odd power of a negative number is negative.
Two terms given
The 3rd term of a geometric sequence is 18 and the 6th term is 486. Find the common ratio and the first term.
Working
Have a go first. Then open the working one line at a time.
-
Two equations
\(u_1 r^2 = 18\) and \(u_1 r^5 = 486\)
Write each term with \(u_n = u_1 r^{n-1}\).
-
Divide them
\(r^3 = \frac{486}{18} = 27\), so \(r = 3\)
Dividing removes \(u_1\).
-
The first term
\(u_1 = \frac{18}{3^2} = 2\)
Put \(r = 3\) back into the first equation.
Two possible ratios
The 2nd term of a geometric sequence is 10 and the 4th term is 250. Find the possible values of the common ratio.
Working
Have a go first. Then open the working one line at a time.
-
Two equations
\(u_1 r = 10\) and \(u_1 r^3 = 250\)
The 2nd term is one multiplication from the first; the 4th is three.
-
Divide them
\(r^2 = 25\), so \(r = 5\) or \(r = -5\)
Two numbers square to 25, so there are two sequences.
-
The first terms
\(r = 5\) gives \(u_1 = 2\). \(r = -5\) gives \(u_1 = -2\).
Check: 2, 10, 50, 250 and −2, 10, −50, 250.
How many terms?
Find the number of terms in the geometric sequence 4, 12, 36, …, 8748.
Working
Have a go first. Then open the working one line at a time.
-
The rule
\(u_n = 4 \times 3^{n-1}\)
First term 4, ratio 3.
-
Set it equal to the last term
\(4 \times 3^{n-1} = 8748\)
From a table on the calculator, or nSolve: \(n = 8\).
A town that grows
A town has 18 000 people. A model predicts the population will grow by 3% each year. Predict the population in 6 years' time.
Working
Have a go first. Then open the working one line at a time.
-
The multiplier
\(1 + \frac{3}{100} = 1.03\)
A 3% rise multiplies by 1.03.
-
After 6 years
\(18\,000 \times 1.03^6 = 21\,492.9\ldots\)
Six years means six multiplications.
-
Round sensibly
About 21 500 people
A count of people from a model: 3 significant figures is enough.
A dose that wears off
A patient has 240 mg of a medicine in their blood. The amount falls by 15% each hour. Find the amount after 5 hours.
Working
Have a go first. Then open the working one line at a time.
-
The multiplier
\(1 - \frac{15}{100} = 0.85\)
A 15% fall multiplies by 0.85.
-
After 5 hours
\(240 \times 0.85^5 = 106.48\ldots\)
About 106 mg.
First term below a value
A patient has 400 mg of a drug in their blood. The amount falls by 20% each hour. After how many complete hours is the amount first below 50 mg?
Working
Have a go first. Then open the working one line at a time.
-
The model
\(u_n = 400 \times 0.8^{n}\)
Falling by 20% multiplies by 0.8 each hour. Here \(n\) is the number of hours.
-
Compare with 50
\(400(0.8)^9 = 53.69\), \(400(0.8)^{10} = 42.95\)
Put the rule in a table, or solve \(400(0.8)^n = 50\) on the calculator: \(n = 9.32\ldots\).
-
Answer
10 hours
The first whole number of hours with the amount below 50 mg.
Show that the ratio is negative
In a geometric sequence, \(u_2 = -12\) and \(u_5 = 96\). (a) Show that the common ratio is negative. (b) Find the common ratio and the first term.
Working
Have a go first. Then open the working one line at a time.
-
Two equations
\(u_1 r = -12\) and \(u_1 r^4 = 96\)
Write each term with \(u_n = u_1 r^{n-1}\).
-
Divide them
\(r^3 = \frac{96}{-12} = -8\)
Dividing removes \(u_1\).
-
(a)
\(r^3 < 0\), so \(r < 0\)
A positive number cubed is positive and a negative number cubed is negative. So the ratio is negative. For "show that", start from the two terms. Checking that an answer works does not earn the marks.
-
(b)
\(r = -2\), \(u_1 = \frac{-12}{-2} = 6\)
Check: 6, −12, 24, −48, 96.