An arithmetic sequence adds a fixed amount each time: 3, 7, 11, 15 adds 4.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Geometric
A geometric sequence multiplies by a fixed factor: 3, 6, 12, 24 doubles.
Two ways to write a rule
Recursive says how to get the next from the last. Explicit gives the nth term directly.
The nth terms
Arithmetic: aₙ = a₁ + (n − 1)d. Geometric: aₙ = a₁ · r^(n−1).
Adding the same, or multiplying by the same
An arithmetic sequence adds a constant difference each term; a geometric sequence multiplies by a constant ratio. Checking differences and then ratios identifies which you have, and often it is neither.
Explicit formulas
Arithmetic: aₙ = a₁ + (n − 1)d. Geometric: aₙ = a₁rⁿ⁻¹. The n − 1 appears because the first term has had no steps applied yet. Getting that off-by-one right is where most errors in the topic live.
Recursive definitions
A recursive formula gives each term from the previous one: aₙ = aₙ₋₁ + d. It is closer to how the sequence is generated but useless for finding the hundredth term directly, which is why explicit formulas matter.
Sequences are functions on the integers
Arithmetic sequences are linear functions restricted to whole-number inputs; geometric ones are exponential functions similarly restricted. Seeing the connection means everything you know about those functions transfers.
Step 2: Try It Yourself
Tap and try it out.
The ringed squares are the multiples of 4. They land in a pattern.
Step 3: Watch an Example
One step at a time.
Watch Ravi Find the 20th Term
The sequence starts 5, 9, 13, 17.
- Step 1
The difference between consecutive terms is 4 each time, so it is arithmetic.
Step 4: Your Turn
Practice makes it stick.
The Tenth Term
Problem 1 of 2
Sequence 2, 5, 8, 11. What is the 10th term?
The Geometric One
Problem 2 of 2
Sequence 2, 6, 18, 54. What is the 5th term?
Find the Term
1 of 4
Sequence 4, 7, 10. What is the 8th term?
2 of 4
Sequence 3, 6, 12. What is the 5th term?
3 of 4
Sequence 100, 90, 80. What is the 6th term?
4 of 4
Sequence 1, 2, 4, 8. Arithmetic or geometric?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Sequence 6, 11, 16. What is the 10th term?
Question 2 of 2
What distinguishes a geometric sequence?
What You Learned
- Arithmetic sequences add a constant; geometric ones multiply by a constant.
- Recursive rules give the next term; explicit rules give the nth directly.
- aₙ = a₁ + (n − 1)d, and aₙ = a₁ · r^(n−1).