A geometric sequence multiplies by the same amount, the common ratio, to get from each term to the next.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Finding the ratio
Divide any term by the one before it. A constant answer confirms the sequence is geometric.
The nth term
aₙ = a₁ · r^(n−1). The same (n − 1) appears, for the same reason as before.
Ratios below 1
A ratio between 0 and 1 shrinks the terms. It is still geometric, just decaying.
Negative ratios
A negative ratio alternates the signs: 2, −6, 18, −54 has ratio −3.
It is an exponential function
Plotted against position, geometric terms sit on an exponential curve, with the ratio as the base.
Multiplying the same amount
A geometric sequence has a constant ratio between consecutive terms. It is an exponential function evaluated at whole numbers, which is why its graph curves.
The explicit formula
aₙ = a₁rⁿ⁻¹. The exponent is n − 1 for the same reason as before — the first term has had no multiplications applied to it yet.
Finding the ratio
Divide any term by the one before it. If different pairs give different answers, the sequence is not geometric, which is a useful test as well as a computation.
When the ratio is under one
A ratio between 0 and 1 makes the terms shrink towards zero without reaching it. Bouncing balls and halving quantities behave this way, and it is the discrete version of exponential decay.
Step 2: Try It Yourself
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Step 3: Watch an Example
One step at a time.
Watch Diego Find the 6th Term
The sequence is 3, 6, 12, 24.
- Step 1
The ratio is 6 ÷ 3 = 2, and it holds for every pair.
Step 4: Your Turn
Practice makes it stick.
The Ratio
Problem 1 of 2
4, 12, 36, 108. What is the common ratio?
The Term
Problem 2 of 2
a₁ = 3 and r = 2. What is the 6th term?
Multiply Along
1 of 8
2, 6, 18, 54. Common ratio?
2 of 8
80, 40, 20, 10. Common ratio?
3 of 8
a₁ = 5, r = 2. What is the 4th term?
4 of 8
a₁ = 1, r = 3. What is the 5th term?
5 of 8
2, −6, 18, −54. Common ratio?
6 of 8
a₁ = 100, r = 0.5. What is the 4th term?
7 of 8
Sort each sequence by its type.
Tap something to move it.
- Empty
- Empty
8 of 8
a₁ = 2, r = 4. What is the 4th term?
Step 5: Quick Check
Show what you know.
Question 1 of 2
a₁ = 4, r = 3. What is the 4th term?
Question 2 of 2
A geometric sequence corresponds to which kind of function?
What You Learned
- A geometric sequence multiplies by a constant common ratio.
- aₙ = a₁ · r^(n−1).
- Arithmetic sequences are linear; geometric sequences are exponential.