Arithmetic Sequence: Complete Guide
An arithmetic sequence (also known as arithmetic progression or AP) is a sequence of numbers where the difference between any two consecutive terms remains constant. This constant value is called the common difference, denoted by \(d\). Arithmetic sequences are fundamental in mathematics and appear extensively in real-world applications including finance, physics, computer science, and engineering.
The general form of an arithmetic sequence is: \(a, a+d, a+2d, a+3d, \ldots\) where \(a\) is the first term and \(d\) is the common difference.
What is an Arithmetic Sequence?
Definition: An arithmetic sequence is an ordered list of numbers in which each term after the first is obtained by adding a fixed constant to the previous term. This fixed constant is the common difference.
Simple Example
Consider the sequence: 3, 7, 11, 15, 19, ...
Analysis:
- First term \(a = 3\)
- Common difference \(d = 7 - 3 = 4\)
- Each term increases by 4
This is an arithmetic sequence because the difference between consecutive terms is constant (always 4).
Common Difference
The common difference is the hallmark feature of an arithmetic sequence. It can be positive, negative, or zero.
Common Difference Formula
\[d = a_n - a_{n-1}\]
Where:
- \(d\) = common difference
- \(a_n\) = any term in the sequence
- \(a_{n-1}\) = the term before \(a_n\)
Positive Common Difference
When \(d > 0\): The sequence is increasing.
Example: 5, 8, 11, 14, 17, ... (\(d = 3\))
Negative Common Difference
When \(d < 0\): The sequence is decreasing.
Example: 20, 15, 10, 5, 0, ... (\(d = -5\))
Zero Common Difference
When \(d = 0\): All terms are equal (constant sequence).
Example: 7, 7, 7, 7, 7, ... (\(d = 0\))
Nth Term of Arithmetic Sequence
One of the most important formulas in arithmetic sequences allows you to find any term in the sequence without listing all previous terms.
General Term (Nth Term) Formula
\[a_n = a_1 + (n-1)d\]
Where:
- \(a_n\) = the nth term (the term you want to find)
- \(a_1\) = the first term of the sequence
- \(n\) = the position of the term in the sequence
- \(d\) = the common difference
💡 Key Insight: The formula \(a_n = a_1 + (n-1)d\) can also be written as \(a_n = a_{n-1} + d\), which shows that each term equals the previous term plus the common difference.
Worked Example: Finding the Nth Term
Problem: Find the 25th term of the arithmetic sequence: 4, 9, 14, 19, ...
Solution:
Step 1: Identify the first term and common difference
- \(a_1 = 4\)
- \(d = 9 - 4 = 5\)
- \(n = 25\)
Step 2: Apply the nth term formula
\[a_{25} = 4 + (25-1) \times 5\]
\[a_{25} = 4 + 24 \times 5\]
\[a_{25} = 4 + 120\]
\[a_{25} = 124\]
Answer: The 25th term is 124.
Sum of Arithmetic Sequence
The sum of the first \(n\) terms of an arithmetic sequence is called an arithmetic series. There are two main formulas for calculating this sum.
Sum Formula 1: Using First Term and Common Difference
\[S_n = \frac{n}{2}[2a_1 + (n-1)d]\]
Use this when: You know the first term, common difference, and number of terms.
Sum Formula 2: Using First and Last Terms
\[S_n = \frac{n}{2}(a_1 + a_n)\]
Use this when: You know the first term, last term, and number of terms.
Where:
- \(S_n\) = sum of the first \(n\) terms
- \(n\) = number of terms to be added
- \(a_1\) = first term
- \(a_n\) = last term (nth term)
- \(d\) = common difference
Worked Example: Finding the Sum
Problem: Find the sum of the first 20 terms of the sequence: 3, 7, 11, 15, ...
Solution Method 1: Using \(S_n = \frac{n}{2}[2a_1 + (n-1)d]\)
- \(a_1 = 3\)
- \(d = 4\)
- \(n = 20\)
\[S_{20} = \frac{20}{2}[2(3) + (20-1)(4)]\]
\[S_{20} = 10[6 + 76]\]
\[S_{20} = 10 \times 82 = 820\]
Solution Method 2: Using \(S_n = \frac{n}{2}(a_1 + a_n)\)
First, find \(a_{20}\): \(a_{20} = 3 + (20-1)(4) = 3 + 76 = 79\)
\[S_{20} = \frac{20}{2}(3 + 79) = 10 \times 82 = 820\]
Answer: The sum of the first 20 terms is 820.
Important Properties of Arithmetic Sequences
Property 1: Middle Term Property
For any three consecutive terms in an arithmetic sequence, the middle term is the average of the other two:
\[a_n = \frac{a_{n-1} + a_{n+1}}{2}\]
Property 2: Linear Relationship
The nth term of an arithmetic sequence is a linear function of \(n\). The graph of \(a_n\) versus \(n\) is always a straight line.
Property 3: Sum Symmetry
The sum of terms equidistant from the beginning and end is constant:
\[a_1 + a_n = a_2 + a_{n-1} = a_3 + a_{n-2} = \ldots\]
Property 4: Arithmetic Mean
If \(a, b, c\) are in arithmetic sequence, then \(b\) is called the arithmetic mean of \(a\) and \(c\):
\[b = \frac{a + c}{2}\]
Real-World Applications
Arithmetic sequences appear frequently in everyday life and various professional fields. Understanding them is essential for practical problem-solving.
1. Financial Planning
Savings Accounts: When making regular equal deposits into a savings account (without interest), the total savings forms an arithmetic sequence. If you deposit $100 every month, your savings after each month form the sequence: 100, 200, 300, 400, ... (\(d = 100\)).
Loan Amortization: Many loan payment schedules follow arithmetic patterns where the principal paid increases arithmetically while interest decreases.
2. Architecture and Construction
Stadium Seating: Rows in stadiums and theaters are often arranged so that the number of seats per row increases arithmetically. For example: Row 1 has 20 seats, Row 2 has 24 seats, Row 3 has 28 seats, etc.
Staircases: The height gained with each step forms an arithmetic sequence when steps are uniform.
3. Time and Motion
Clock Movements: The position of clock hands moves in arithmetic progression. A second hand moves 6 degrees every second (0°, 6°, 12°, 18°, ...).
Uniform Motion: When an object moves at constant speed, the distance traveled at equal time intervals forms an arithmetic sequence.
4. Calendar and Time
Years and Dates: Years progress arithmetically (2020, 2021, 2022, ...). Leap years occur in arithmetic sequence: 2020, 2024, 2028, 2032, ... (\(d = 4\)).
5. Manufacturing and Production
Production Planning: If a factory increases production by a fixed amount each day, the total production follows an arithmetic sequence.
6. Computer Science
Array Indexing: Array indices in programming form arithmetic sequences (0, 1, 2, 3, ...).
Algorithm Analysis: Certain algorithms have time complexities that involve arithmetic series calculations.
Quick Reference: Essential Formulas
| Formula | Expression | Use Case |
|---|---|---|
| Common Difference | \(d = a_n - a_{n-1}\) | Finding the constant difference between terms |
| Nth Term | \(a_n = a_1 + (n-1)d\) | Finding any specific term in the sequence |
| Recursive Formula | \(a_n = a_{n-1} + d\) | Finding the next term from the previous term |
| Sum (Method 1) | \(S_n = \frac{n}{2}[2a_1 + (n-1)d]\) | Sum when you know first term and common difference |
| Sum (Method 2) | \(S_n = \frac{n}{2}(a_1 + a_n)\) | Sum when you know first and last terms |
| Arithmetic Mean | \(b = \frac{a + c}{2}\) | Finding the middle term between two terms |
Practice Problems
Solution:
- First term: \(a_1 = 7\)
- Common difference: \(d = 13 - 7 = 6\)
- Position: \(n = 50\)
Using the formula: \(a_n = a_1 + (n-1)d\)
\(a_{50} = 7 + (50-1)(6) = 7 + 294 = 301\)
Answer: 301
Solution:
- First term: \(a_1 = 2\)
- Common difference: \(d = 3\)
- Number of terms: \(n = 15\)
Using the formula: \(S_n = \frac{n}{2}[2a_1 + (n-1)d]\)
\(S_{15} = \frac{15}{2}[2(2) + (15-1)(3)]\)
\(S_{15} = \frac{15}{2}[4 + 42] = \frac{15}{2}(46) = 15 \times 23 = 345\)
Answer: 345
Solution:
We know: \(a_n = a_1 + (n-1)d\)
For \(n=5\): \(17 = a_1 + 4d\) ... (Equation 1)
For \(n=9\): \(33 = a_1 + 8d\) ... (Equation 2)
Subtract Equation 1 from Equation 2:
\(33 - 17 = (a_1 + 8d) - (a_1 + 4d)\)
\(16 = 4d\)
\(d = 4\)
Substitute \(d = 4\) into Equation 1:
\(17 = a_1 + 4(4)\)
\(17 = a_1 + 16\)
\(a_1 = 1\)
Answer: \(a_1 = 1\), \(d = 4\)
Key Terminology
| Term | Definition |
|---|---|
| Arithmetic Sequence | A sequence where consecutive terms differ by a constant value |
| Common Difference (d) | The constant value added to each term to get the next term |
| First Term (a₁ or a) | The initial term of the sequence |
| Nth Term (aₙ) | The term at position n in the sequence |
| Arithmetic Series | The sum of terms in an arithmetic sequence |
| Arithmetic Mean | The average of two numbers; the middle term in an arithmetic sequence |
Tips and Strategies for Success
✓ Tip 1: Always identify \(a_1\) and \(d\) first before using any formula.
✓ Tip 2: Check if the sequence is actually arithmetic by verifying that the difference between consecutive terms is constant.
✓ Tip 3: For sum problems, choose the formula based on what information you have. If you know the last term, use \(S_n = \frac{n}{2}(a_1 + a_n)\) as it's usually faster.
✓ Tip 4: Remember that \(d\) can be negative (decreasing sequence) or zero (constant sequence).
✓ Tip 5: When finding which term has a specific value, set up the equation \(a_n = a_1 + (n-1)d\) and solve for \(n\).
Common Mistakes to Avoid
❌ Mistake 1: Confusing (n-1) with n
Wrong: \(a_n = a_1 + nd\)
Correct: \(a_n = a_1 + (n-1)d\)
Reason: There are (n-1) steps from the first term to the nth term.
❌ Mistake 2: Forgetting to check if it's arithmetic
Always verify that the difference between consecutive terms is constant before applying arithmetic sequence formulas.
❌ Mistake 3: Sign errors with negative common difference
When \(d\) is negative, be careful with subtraction. For example, if \(d = -3\), then \(a_1 + (n-1)d = a_1 + (n-1)(-3) = a_1 - 3(n-1)\).
Interesting Mathematical Facts
🎓 Historical Fact: The famous mathematician Carl Friedrich Gauss, at age 10, amazed his teacher by quickly summing the integers from 1 to 100. He recognized it as an arithmetic sequence and used the formula \(S_n = \frac{n(a_1 + a_n)}{2}\) to get 5050 instantly!
🎓 Connection to Linear Functions: Every arithmetic sequence can be represented as a linear function \(f(n) = a_1 + (n-1)d\), where the slope is the common difference \(d\).
🎓 Curriculum Relevance: Arithmetic sequences are fundamental in IB Mathematics, AP Calculus prerequisites, GCSE/IGCSE Higher Tier, and appear in SAT/ACT math sections.
🎓 Connection to Other Topics: Arithmetic sequences form the foundation for understanding series, sigma notation, and are essential for studying geometric sequences, recursion, and calculus concepts like sequences and limits.
Summary
Arithmetic sequences are fundamental mathematical structures characterized by a constant difference between consecutive terms. Mastering the key formulas—the nth term formula \(a_n = a_1 + (n-1)d\) and the sum formulas—enables you to solve a wide variety of problems in mathematics and real-world applications.
Whether you're calculating savings growth, analyzing patterns, or preparing for standardized tests like the IB, AP, SAT, or GCSE exams, understanding arithmetic sequences provides essential problem-solving tools that extend far beyond the classroom.
🎯 Key Takeaway: Always identify the first term (\(a_1\)) and common difference (\(d\)) first, then choose the appropriate formula based on what you need to find: a specific term, or the sum of multiple terms.
About the Author
Adam
Co-Founder @RevisionTown
Math Expert specializing in various international curricula including IB (International Baccalaureate), AP (Advanced Placement), GCSE, IGCSE, A-Levels, and more. Dedicated to creating comprehensive, accessible educational resources that help students master mathematics across all levels and examination boards.