A geometric sequence is a list of numbers where each term changes by multiplying or dividing by the same nonzero value, called the common ratio, or r. But learning how to find r in a geometric sequence is one of the most important skills for working with exponential growth, exponential decay, financial formulas, population models, and many other real-world situations. The common ratio tells you how quickly the sequence grows or shrinks from one term to the next Which is the point..
Worth pausing on this one Easy to understand, harder to ignore..
Introduction to Geometric Sequences
A geometric sequence is a sequence in which every term after the first is found by multiplying the previous term by a fixed number. That fixed number is called the common ratio, written as r.
For example:
[ 3,\ 6,\ 12,\ 24,\ 48,\ldots ]
In this sequence, each term is multiplied by 2 to get the next term:
[ 3 \times 2 = 6,\quad 6 \times 2 = 12,\quad 12 \times 2 = 24 ]
So the common ratio is:
[ r = 2 ]
The common ratio is what makes a sequence geometric. If the ratio between consecutive terms is always the same, then the sequence is geometric.
What Does r Mean in a Geometric Sequence?
The value of r tells you the pattern of the sequence.
If:
[ r > 1 ]
then the sequence grows rapidly Surprisingly effective..
Example:
[ 5,\ 10,\ 20,\ 40,\ldots ]
Here, (r = 2), so each term doubles.
If:
[ 0 < r < 1 ]
then the sequence decreases toward zero Turns out it matters..
Example:
[ 80,\ 40,\ 20,\ 10,\ldots ]
Here, (r = \frac{1}{2}), so each term is multiplied by one-half That's the part that actually makes a difference..
If:
[ r = 1 ]
then every term is the same Most people skip this — try not to..
Example:
[ 7,\ 7,\ 7,\ 7,\ldots ]
Here, (r = 1) That's the part that actually makes a difference..
If:
[ r < 0 ]
then the sequence alternates signs.
Example:
[ 4,\ -8,\ 16,\ -32,\ldots ]
Here, (r = -2). Each term is multiplied by negative 2.
The Basic Formula for Finding r
The simplest way to find r in a geometric sequence is to divide any term by the term that comes immediately before it.
If you have two consecutive terms, use:
[ r = \frac{a_n}{a_{n-1}} ]
where:
- (a_n) is the later term
- (a_{n-1}) is the term before it
Here's one way to look at it: suppose the sequence is:
[ 6,\ 18,\ 54,\ 162,\ldots ]
To find (r), divide 18 by 6:
[ r = \frac{18}{6} = 3 ]
Check another pair of terms:
[ \frac{54}{18} = 3 ]
[ \frac{162}{54} = 3 ]
Since the ratio is the same, the common ratio is:
[ r = 3 ]
Example 1: Finding r with Whole Numbers
Find the common ratio in the sequence:
[ 4,\ 12,\ 36,\ 108,\ldots ]
Divide the second term by the first term:
[ r = \frac{12}{4} = 3 ]
Check:
[ \frac{36}{12} = 3 ]
[ \frac{108}{36} = 3 ]
So the common ratio is:
[ \boxed{r = 3} ]
This means each term is three times the previous term.
Example 2: Finding r with Fractions
Find the common ratio in the sequence:
[ \frac{3}{4},\ \frac{9}{8},\ \frac{27}{16},\ \frac{81}{32},\ldots ]
Divide the second term by the first term:
[ r = \frac{\frac{9}{8}}{\frac{3}{4}} ]
To divide fractions, multiply by the reciprocal:
[ r = \frac{9}{8} \times \frac{4}{3} ]
[ r = \frac{36}{24} ]
[ r = \frac{3}{2} ]
So the common ratio is:
[ \boxed{r = \frac{3}{2}} ]
This means each term is multiplied by 1.5 to get the next term.
Example 3: Finding r with Decimals
Find the common ratio in the sequence:
[ 0.5,\ 1,\ 2,\ 4,\ldots ]
Divide the second term by the first:
[ r = \frac{1}{0.5} = 2 ]
Check:
[ \frac{2}{1} = 2 ]
[ \frac{4}{2} = 2 ]
So:
[ \boxed{r = 2} ]
Finding r When Only Two Terms Are Given
Sometimes you may not be given consecutive terms. To give you an idea, you might be given the first term and the fifth term. In that case, you can use the formula for the (n)th term of a geometric sequence:
[ a_n = a_1r^{n-1} ]
where:
- (a_n) is the (n)th term
- (a_1) is the first term
- (r) is the common ratio
- (n) is the position of the term
Suppose:
[ a_1 = 5 ]
and:
[ a_4 = 135 ]
Use the formula:
[ a_n = a_1r^{n-1} ]
Substitute the values:
[ 135 = 5r^{4-1} ]
[ 135 = 5r^3 ]
Divide both sides by 5:
[ 27 = r^3 ]
Take the cube root:
[ r = 3 ]
So the common ratio is:
[ \boxed{r = 3} ]
Finding r When the Terms Are Not Consecutive
If the terms are not next to each other, count how many jumps it takes to get from one term to the other But it adds up..
Here's one way to look at it: suppose:
[ a_2 = 10 ]
