Arithmetic Sequences
Explore sequences with a constant common difference, learn the nth term formula, and solve real-world applications.
What Is an Arithmetic Sequence?
An arithmetic sequence (also called an arithmetic progression, or AP) is a sequence where each term is obtained by adding a constant value to the previous term. This constant is called the common difference (d).
Examples
The nth Term Formula
To find any term in an arithmetic sequence without listing all previous terms, use the nth term formula:
Tn = a + (n - 1)d
where a = first term, d = common difference, n = term number
For example, in the sequence 5, 8, 11, 14, ...: a = 5, d = 3. The 20th term is T20 = 5 + (20 - 1)(3) = 5 + 57 = 62.
Arithmetic Mean and Applications
The arithmetic mean of two numbers a and b is (a + b)/2. In a sequence, the middle term between two terms is their arithmetic mean.
Salary Increases
A starting salary of $50,000 with annual raises of $3,000 forms an AP: 50000, 53000, 56000, ...
Seating Arrangements
A theatre with 20 seats in the front row and 2 extra seats per row is an AP: 20, 22, 24, ...
Saving Plans
Saving $100 more each month than the last: $200, $300, $400, ... is an AP with d = 100.
Depreciation
An asset losing $5,000 in value each year forms a decreasing AP with d = -5000.
Key Vocabulary
Common Difference (d)
The constant value added to each term to get the next term. Found by d = Tn+1 - Tn.
First Term (a)
The starting term of the sequence, also written as T1. It anchors the entire sequence.
nth Term (Tn)
The general term formula Tn = a + (n - 1)d gives any term in the sequence directly.
Arithmetic Mean
The average of two values: (a + b)/2. The middle term between two terms in an AP.
Worked Examples
Find the 15th term of the sequence 2, 9, 16, 23, ...
Step 1: Identify a = 2, d = 9 - 2 = 7.
Step 2: Tn = a + (n - 1)d = 2 + (15 - 1)(7) = 2 + 98 = 100.
Answer: T15 = 100.
The 5th term of an AP is 23 and the 12th term is 58. Find a and d.
Step 1: T5 = a + 4d = 23 and T12 = a + 11d = 58.
Step 2: Subtract: 7d = 35, so d = 5.
Step 3: a + 4(5) = 23, so a = 23 - 20 = 3.
Answer: a = 3, d = 5. The sequence is 3, 8, 13, 18, 23, ...
Which term of the sequence 4, 11, 18, 25, ... is equal to 200?
Step 1: a = 4, d = 7. Set Tn = 200.
Step 2: 4 + (n - 1)(7) = 200, so 7(n - 1) = 196, giving n - 1 = 28.
Answer: n = 29. The 29th term equals 200.
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is the common difference of the sequence 10, 7, 4, 1, -2, ...?
Question 2
Find the 10th term of the AP: 5, 11, 17, 23, ...
Question 3
The arithmetic mean of 8 and 22 is:
Question 4
If T1 = 3 and d = 5, what is Tn in terms of n?
Question 5
Is the sequence 1, 4, 9, 16, 25, ... an arithmetic sequence?
Key Concepts Summary
- ● An arithmetic sequence has a constant common difference d between terms.
- ● The nth term formula is Tn = a + (n - 1)d.
- ● d can be positive (increasing), negative (decreasing), or zero (constant).
- ● The arithmetic mean of two values a and b is (a + b)/2.
- ● Arithmetic sequences model linear growth in real-world situations.