How to Find the Common Difference Formula with Solved Examples
The concept of common difference is essential in mathematics and helps in solving real-world and exam-level problems efficiently. It forms the backbone of arithmetic sequences and is a must-know concept for school exams and board preparations.
Understanding Common Difference
A common difference refers to the fixed value by which each term in an arithmetic sequence increases or decreases from the previous term. This concept is widely used in arithmetic progression, sequences and series, and number pattern analysis. In every arithmetic sequence, the same amount (positive or negative) is added or subtracted each time, making the pattern easy to recognise and work with.
Formula Used in Common Difference
The standard formula for finding the common difference in an arithmetic sequence is:
\( d = a_{n} - a_{n-1} \)
where \( a_{n} \) is any term and \( a_{n-1} \) is its previous term.
Here’s a helpful table to understand common difference more clearly with a few examples:
Common Difference Table
| Sequence | Common Difference (d) | Is AP? |
|---|---|---|
| 2, 5, 8, 11 | 3 | Yes |
| 10, 8, 6, 4 | -2 | Yes |
| 4, 7, 12, 19 | Not constant | No |
| 1, 1, 1, 1 | 0 | Yes |
This table shows how the pattern of common difference appears regularly in real cases and helps decide if a sequence is arithmetic.
How to Find the Common Difference
Finding the common difference in a sequence is simple if you follow these steps:
1. Write out the given arithmetic sequence.
2. Pick any two consecutive terms—let's say the second and first term.
3. Subtract the earlier term from the later term using the formula \( d = a_{n} - a_{n-1} \).
4. If you get the same result for every consecutive pair, that value is your common difference.
Worked Example – Solving a Problem
Let’s see a step-by-step example to make things clearer:
Example: What is the common difference in the sequence 7, 12, 17, 22?
1. Write out the first two terms: 12 (second term), 7 (first term).
2. Subtract: 12 - 7 = 5
3. Take the next two terms: 17 (third term), 12 (second term).
4. Subtract: 17 - 12 = 5
5. Since the difference is the same for all pairs, the common difference (d) = 5.
Practice Problems
- Find the common difference in the sequence 15, 11, 7, 3.
- If the first term is 6 and the second term is 10, what is the common difference?
- Does the sequence 20, 30, 41, 51 have a common difference? If so, what is it?
- The 5th term of an AP is 20 and the 6th term is 27. Find the common difference.
- Is the sequence 0, 0, 0 an arithmetic progression? What is the common difference?
Common Mistakes to Avoid
- Mixing up the common difference with the common ratio of geometric sequences.
- Subtracting in the wrong order (always subtract the previous term from the current).
- Assuming any sequence has a common difference—it only applies to arithmetic sequences.
Real-World Applications
The concept of common difference appears in areas such as payment plans (equal installments), cricket score increments, patterns in tiling, daily routines, and more. Vedantu helps students see how maths applies beyond the classroom, especially when working with APs and sequences in real life.
Related Concepts and Next Steps
- Learn about the nth term of AP to find any term quickly using the common difference.
- Explore the differences between arithmetic and geometric patterns in arithmetic and geometric sequences.
- Practice finding sums using sum of n terms formulae.
We explored the idea of common difference, how to apply it, solve related problems, and understand its real-life relevance. Practice more with Vedantu to build confidence in these concepts.
FAQs on Common Difference in Arithmetic Progression Explained
1. What is the common difference in an arithmetic sequence?
The common difference is the fixed number that is added or subtracted between consecutive terms in an arithmetic sequence. In an arithmetic sequence, each term increases or decreases by the same constant value.
- If the sequence is 2, 5, 8, 11, the common difference is 3.
- If the sequence is 10, 7, 4, 1, the common difference is -3.
2. How do you find the common difference in an arithmetic sequence?
The common difference is found by subtracting any term from the term that comes after it. The formula is d = a₂ − a₁.
- Step 1: Identify two consecutive terms.
- Step 2: Subtract the first from the second.
- Example: In 4, 9, 14, 19 → d = 9 − 4 = 5.
3. What is the formula for common difference?
The formula for common difference is d = aₙ − aₙ₋₁, where aₙ is any term and aₙ₋₁ is the previous term. It can also be written as d = a₂ − a₁ for the first two terms.
- This formula works only for arithmetic sequences.
- The value of d remains constant throughout the sequence.
4. What does a negative common difference mean?
A negative common difference means the arithmetic sequence is decreasing. Each new term is obtained by subtracting a fixed number.
- Example: 15, 11, 7, 3 → d = 11 − 15 = -4.
- The terms decrease by 4 each time.
5. How is common difference used in the nth term formula?
The common difference is used in the nth term formula of an arithmetic sequence: aₙ = a₁ + (n − 1)d. Here:
- a₁ = first term
- d = common difference
- n = term number
6. Can the common difference be zero?
Yes, the common difference can be zero, which means all terms in the arithmetic sequence are equal. If d = 0, every term remains constant.
- Example: 7, 7, 7, 7 → d = 0.
- This is called a constant sequence.
7. What is the difference between common difference and common ratio?
The common difference is used in arithmetic sequences, while the common ratio is used in geometric sequences. Key differences:
- Common difference (d): Add or subtract a fixed number.
- Common ratio (r): Multiply or divide by a fixed number.
- Example arithmetic: 2, 5, 8 → d = 3.
- Example geometric: 2, 6, 18 → r = 3.
8. How do you find the common difference if you know the first term and nth term?
You can find the common difference using the formula d = (aₙ − a₁) / (n − 1). Steps:
- Step 1: Subtract the first term from the nth term.
- Step 2: Divide by (n − 1).
- Example: If a₁ = 2, a₅ = 18 → d = (18 − 2) / 4 = 4.
9. Why is the common difference important in arithmetic sequences?
The common difference is important because it defines the pattern and growth of an arithmetic sequence. It helps to:
- Identify whether the sequence is increasing or decreasing.
- Find any term using aₙ = a₁ + (n − 1)d.
- Calculate the sum using arithmetic series formulas.
10. Can you give a real-life example of common difference?
A real-life example of common difference is saving a fixed amount of money each week. If you save $50 more each week than the previous week, the common difference is 50.
- Week 1: $100
- Week 2: $150
- Week 3: $200
