An arithmetic sequence adds the same amount each step — the common difference.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Geometric
A geometric sequence multiplies by the same amount each step — the common ratio.
The nth term, arithmetic
aₙ = a₁ + (n − 1)d. The n − 1 is there because the first term takes no steps.
The nth term, geometric
aₙ = a₁ · r^(n−1), for exactly the same reason.
Adding or multiplying
Arithmetic sequences add a constant difference; geometric sequences multiply by a constant ratio. Checking differences and then ratios identifies which you have, and confirms when it is neither.
The explicit formulas
aₙ = a₁ + (n − 1)d for arithmetic, aₙ = a₁rⁿ⁻¹ for geometric. The n − 1 appears because no step has been applied to the first term. That off-by-one is where most errors in the topic occur.
The sum formulas
Arithmetic: n times the average of first and last. Geometric: a₁(1 − rⁿ)/(1 − r). The first comes from pairing terms from the ends, the second from subtracting r times the sum from the sum.
Where sequences lead
An infinite geometric series converges when |r| < 1. Convergence is a limit question, which is why sequences sit next to the limits chapter — the entire idea of an infinite sum depends on limits.
Step 2: Try It Yourself
Tap and try it out.
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Step 3: Watch an Example
One step at a time.
Watch Yusuf Find the Tenth Term
The sequence starts 4, 7, 10, 13.
- Step 1
Each step adds 3, so it is arithmetic with d = 3.
Step 4: Your Turn
Practice makes it stick.
The Difference
Problem 1 of 2
5, 9, 13, 17. What is the common difference?
The Ratio
Problem 2 of 2
3, 6, 12, 24. What is the common ratio?
Which Kind, and Which Term
1 of 8
2, 5, 8, 11. Common difference?
2 of 8
2, 5, 8, 11. What is the 5th term?
3 of 8
1, 3, 9, 27. Common ratio?
4 of 8
1, 3, 9, 27. What is the 5th term?
5 of 8
Sort each sequence by its kind.
Tap something to move it.
- Empty
- Empty
6 of 8
a₁ = 6, d = 5. What is a₄?
7 of 8
a₁ = 2, r = 4. What is a₃?
8 of 8
For the nth term, how many steps are taken from the first term?
Step 5: Quick Check
Show what you know.
Question 1 of 1
a₁ = 3, d = 7. What is a₅?
What You Learned
- Arithmetic sequences add a common difference; geometric ones multiply by a common ratio.
- The nth term uses n − 1 steps, because the first term takes none.
- Which kind it is decides which formula applies.