An arithmetic sequence adds the same amount, the common difference, to get from each term to the next.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Finding the difference
Subtract any term from the one after it. If the answer varies, the sequence is not arithmetic.
The nth term
aₙ = a₁ + (n − 1)d. The (n − 1) matters: reaching the 5th term takes four steps, not five.
The off-by-one trap
Using n instead of n − 1 gives an answer exactly one difference too large. It is the commonest error here.
It is a linear function
Plotting term number against value gives points on a line, with the common difference as the slope.
One difference from a line
A sequence exists only at whole-number positions, so the graph is dots rather than a continuous line.
Adding the same amount
An arithmetic sequence has a constant difference between consecutive terms. It is a linear function evaluated only at whole numbers, which is why its graph is a set of points on a line.
The explicit formula
aₙ = a₁ + (n − 1)d. The n − 1 appears because no step has yet been applied to the first term. That off-by-one is where nearly all errors in the topic occur.
The common difference is a slope
Plotting term number against value gives points on a line whose slope is the common difference. Seeing the connection means everything known about linear functions transfers directly.
Finding the difference from any two terms
If two non-consecutive terms are known, divide the difference in values by the difference in positions. That is the slope formula, applied to a sequence.
Step 2: Try It Yourself
Tap and try it out.
The slope is 4: for every 1 across, the line goes 4 up.
Step 3: Watch an Example
One step at a time.
Watch Rosa Find the 20th Term
The sequence is 3, 7, 11, 15.
- Step 1
The common difference is 7 − 3 = 4, and it holds for every pair.
Step 4: Your Turn
Practice makes it stick.
The Difference
Problem 1 of 2
5, 12, 19, 26. What is the common difference?
The Term
Problem 2 of 2
a₁ = 3 and d = 4. What is the 20th term?
Step by Step
1 of 8
2, 5, 8, 11. Common difference?
2 of 8
20, 15, 10, 5. Common difference?
3 of 8
a₁ = 4, d = 3. What is the 10th term?
4 of 8
a₁ = 100, d = −7. What is the 5th term?
5 of 8
To reach the 12th term, how many steps from the first?
6 of 8
a₁ = 6, d = 5. What is the 8th term?
7 of 8
Sort each sequence by whether it is arithmetic.
Tap something to move it.
- Empty
- Empty
8 of 8
a₁ = 2, d = 6. What is the 15th term?
Step 5: Quick Check
Show what you know.
Question 1 of 2
a₁ = 5, d = 4. What is the 10th term?
Question 2 of 2
Why does the formula use n − 1 rather than n?
What You Learned
- An arithmetic sequence adds a constant common difference.
- aₙ = a₁ + (n − 1)d.
- Plotted against position, the terms lie on a line whose slope is d.