A geometric sequence is a sequence in which each term is obtained by multiplying the previous term by a constant number. That constant is called the common ratio. Also, learning to find the common ratio in a geometric sequence is essential for identifying the pattern, writing an explicit formula, calculating missing terms, and solving many algebra problems. The most reliable method is to divide any term by the term immediately before it Still holds up..
What Is the Common Ratio in a Geometric Sequence?
A geometric sequence follows the general form:
[ a_1,\ a_1r,\ a_1r^2,\ a_1r^3,\ldots ]
In this expression:
- (a_1) represents the first term.
- (r) represents the common ratio.
- (a_n) represents the (n)-th term.
The common ratio can be found using the formula:
[ r=\frac{a_{n+1}}{a_n} ]
Simply put, divide a term by the term that comes immediately before it Surprisingly effective..
To give you an idea, consider the geometric sequence:
[ 2,\ 6,\ 18,\ 54 ]
Divide the second term by the first:
[ r=\frac{6}{2}=3 ]
The common ratio is (3). You can verify the answer by dividing the remaining consecutive terms:
[ \frac{18}{6}=3 \qquad\text{and}\qquad \frac{54}{18}=3 ]
Because the ratio remains constant, the sequence is geometric.
Why Finding the Common Ratio Matters
The common ratio reveals the pattern behind a geometric sequence. Once it is known, you can:
- Find any term in the sequence.
- Calculate a missing term.
- Write an explicit formula for the sequence.
- Determine whether a sequence is geometric.
- Solve problems involving exponential growth or decay.
The general formula for the (n)-th term of a geometric sequence is:
[ a_n=a_1r^{n-1} ]
Knowing (r) allows you to extend the sequence as far as necessary. To give you an idea, if the first term is (4) and the common ratio is (3), the sequence begins:
[ 4,\ 12,\ 36,\ 108,\ 324,\ldots ]
Method 1: Divide Consecutive Terms
This is the simplest and most commonly used method.
Step 1: Identify Two Consecutive Terms
Choose a term and the term immediately after it. For the sequence:
[ 5,\ 20,\ 80,\ 320 ]
The first two terms are (5) and (20) Not complicated — just consistent..
Step 2: Divide the Later Term by the Earlier Term
[ r=\frac{20}{5}=4 ]
Step 3: Check the Remaining Terms
[ \frac{80}{20}=4 \qquad\text{and}\qquad \frac{320}{80}=4 ]
The common ratio is (4) Practical, not theoretical..
You only need one pair of consecutive terms if you already know the sequence is geometric. On the flip side, checking another pair helps confirm that the ratio is constant And that's really what it comes down to. No workaround needed..
Example With a Fractional Ratio
Consider:
[ 24,\ 12,\ 6,\ 3 ]
Calculate the ratio:
[ r=\frac{12}{24}=\frac{1}{2} ]
The common ratio is (\frac{1}{2}). The sequence decreases because each term is multiplied by a fraction between zero and one Simple as that..
Example With a Negative Ratio
Consider:
[ 3,\ -6,\ 12,\ -24 ]
The ratio is:
[ r=\frac{-6}{3}=-2 ]
The common ratio is (-2). A negative ratio causes the signs to alternate: positive, negative, positive, negative, and so on Less friction, more output..
Method 2: Use the General Formula
If the first term and a later term are known but no consecutive pair is provided, use:
[ a_n=a_1r^{n-1} ]
Rearrange the formula to solve for (r):
[ r=\sqrt[n-1]{\frac{a_n}{a_1}} ]
Example 1: Finding the Ratio From the First and Fifth Terms
Suppose the first term of a geometric sequence is (2), and the fifth term is (162). Then:
[ 162=2r^4 ]
Divide both sides by (2):
[ 81=r^4 ]
Therefore:
[ r=3 ]
The complete sequence is:
[ 2,\ 6,\ 18,\ 54,\ 162 ]
Example 2: Finding the Ratio From the First and Fourth Terms
Suppose (a_1=1) and (a_4=64). Using the formula:
[ 64=1r^3 ]
Therefore:
[
