Common Difference of an Arithmetic Progression
The common difference is the defining characteristic of an arithmetic progression (AP). It is the constant value by which each term of the sequence increases (or decreases) from the previous term. Understanding the common difference is essential because it is the single parameter, along with the first term, that completely determines an arithmetic progression. Every formula in the AP chapter — the nth term formula, the sum formula, properties of terms — relies on the common difference. For CBSE Class 10 students, the common difference is the very first concept introduced in the chapter on Arithmetic Progressions, and it forms the basis for all subsequent work. Despite its apparent simplicity, the concept requires careful attention: the common difference can be positive, negative, or zero, and finding it correctly from various representations of an AP is a skill that students must master. The common difference also connects to broader mathematical ideas. It is essentially the slope of the linear function that generates the AP terms. If you plot the term values against their position numbers, the common difference is the slope of the resulting straight line. This connection to linear functions makes APs a bridge between sequences and coordinate geometry. This topic provides a thorough exploration of the common difference — its definition, computation, properties, significance, and application through a comprehensive set of examples that cover every type of question students may encounter in CBSE board examinations.
What is Common Difference of AP?
An Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference, denoted by d.
d = an+1 - an (for any term in the AP)
Here, an denotes the nth term and an+1 denotes the next (n+1)th term. The common difference d is the same for every pair of consecutive terms in the AP. This constancy is what distinguishes an arithmetic progression from other types of sequences. In a geometric progression, by contrast, the ratio (not difference) between consecutive terms is constant.
If the first term of the AP is a, then the AP can be written as:
a, a + d, a + 2d, a + 3d, a + 4d, ...
Properties of the Common Difference:
- d > 0: The AP is increasing. Each term is greater than the previous one. Example: 3, 7, 11, 15, ... (d = 4).
- d < 0: The AP is decreasing. Each term is less than the previous one. Example: 20, 15, 10, 5, ... (d = -5).
- d = 0: All terms are equal. The sequence is constant. Example: 5, 5, 5, 5, ... (d = 0).
The common difference can be any real number — integer, fraction, decimal, or irrational number. It is always computed as second term minus first term (or any term minus the immediately preceding term), and this value must be the same throughout the sequence for it to qualify as an AP. If even one pair of consecutive terms gives a different difference, the sequence is not an AP. This is the fundamental test for identifying arithmetic progressions.
Common Difference of an Arithmetic Progression Formula
Formula for Common Difference:
d = a2 - a1 = a3 - a2 = a4 - a3 = ... = an+1 - an
In other words, d equals any term minus the previous term.
Relation to Other AP Formulae:
| Formula | Equation | Role of d |
|---|---|---|
| nth term | an = a + (n - 1)d | d determines the growth rate |
| Sum of n terms | Sn = n/2 [2a + (n - 1)d] | d affects the total sum |
| Common difference from two known terms | d = (am - an) / (m - n) | Finding d when non-consecutive terms are known |
Finding d When Two Non-Consecutive Terms Are Known:
If the mth term am and nth term an are known (m ≠ n), then:
d = (am - an) / (m - n)
This follows from am = a + (m-1)d and an = a + (n-1)d, so am - an = (m-n)d.
Methods
Methods for Finding the Common Difference:
Method 1: Direct Subtraction of Consecutive Terms
If consecutive terms of the AP are given, simply subtract any term from the next term: d = a2 - a1.
Example: For the AP 8, 14, 20, 26, ..., d = 14 - 8 = 6.
Method 2: Using Non-Consecutive Terms
If the mth and nth terms are given, use d = (am - an) / (m - n).
Example: If the 5th term is 23 and the 12th term is 58, then d = (58 - 23)/(12 - 5) = 35/7 = 5.
Method 3: Using the nth Term Formula
If a formula for an is given (like an = 3n + 2), compute d = a2 - a1, or observe that if an is a linear function of n (of the form pn + q), then d = p (the coefficient of n).
Method 4: Checking Whether a Sequence Is an AP
Compute a2 - a1, a3 - a2, a4 - a3, etc. If all these differences are equal, the sequence is an AP with that common difference. If any two differences are unequal, the sequence is NOT an AP.
Common Pitfalls:
- Computing a1 - a2 instead of a2 - a1. Remember: d = next term minus current term.
- Assuming d must be positive. The common difference can be negative (decreasing AP) or zero (constant sequence).
- Not checking all consecutive differences when verifying if a sequence is an AP.
Solved Examples
Example 1: Finding d from a Simple AP
Problem: Find the common difference of the AP: 5, 9, 13, 17, 21, ...
Solution:
d = a2 - a1 = 9 - 5 = 4
Verification: a3 - a2 = 13 - 9 = 4, a4 - a3 = 17 - 13 = 4, a5 - a4 = 21 - 17 = 4. All equal. ✓
Answer: d = 4. This is an increasing AP.
Example 2: Finding d for a Decreasing AP
Problem: Find the common difference: 100, 93, 86, 79, 72, ...
Solution:
d = 93 - 100 = -7
Check: 86 - 93 = -7, 79 - 86 = -7, 72 - 79 = -7. All equal. ✓
Answer: d = -7. This is a decreasing AP — each term is 7 less than the previous.
Example 3: Finding d for a Constant Sequence
Problem: Is the sequence 4, 4, 4, 4 an AP? If so, find d.
Solution:
d = 4 - 4 = 0. Check: all differences are 0.
Yes, this is an AP with d = 0. A constant sequence is a valid AP.
Answer: d = 0.
Example 4: Finding d with Fractional Terms
Problem: Find the common difference of the AP: 1/3, 5/6, 4/3, 11/6, ...
Solution:
d = 5/6 - 1/3 = 5/6 - 2/6 = 3/6 = 1/2
Check: 4/3 - 5/6 = 8/6 - 5/6 = 3/6 = 1/2 ✓
11/6 - 4/3 = 11/6 - 8/6 = 3/6 = 1/2 ✓
Answer: d = 1/2.
Example 5: Finding d from Non-Consecutive Terms
Problem: The 4th term of an AP is 17 and the 9th term is 37. Find the common difference.
Solution:
Using d = (am - an) / (m - n):
d = (a9 - a4) / (9 - 4) = (37 - 17) / 5 = 20/5 = 4
Answer: d = 4.
Example 6: Finding d and the First Term
Problem: The 3rd term of an AP is 12 and the 7th term is 28. Find a and d.
Solution:
a3 = a + 2d = 12 ... (i)
a7 = a + 6d = 28 ... (ii)
Subtracting (i) from (ii): 4d = 16, so d = 4.
From (i): a + 8 = 12, so a = 4.
The AP is: 4, 8, 12, 16, 20, 24, 28, ...
Answer: a = 4, d = 4.
Example 7: Checking if a Sequence is an AP
Problem: Is the sequence 3, 7, 12, 18, 25 an AP?
Solution:
a2 - a1 = 7 - 3 = 4
a3 - a2 = 12 - 7 = 5
Since 4 ≠ 5, the differences are not constant.
Answer: No, the sequence is NOT an AP. (It is actually a sequence of differences 4, 5, 6, 7 — an AP of differences, making it a quadratic sequence.)
Example 8: Finding d Given a Formula for a_n
Problem: If an = 4n - 3, find the common difference.
Solution:
Method 1: a1 = 4(1) - 3 = 1, a2 = 4(2) - 3 = 5. So d = 5 - 1 = 4.
Method 2: Since an = 4n - 3 is linear in n, the coefficient of n gives d = 4.
The AP is: 1, 5, 9, 13, 17, ...
Answer: d = 4.
Example 9: Finding d with Negative and Positive Terms
Problem: Find d for the AP: -8, -3, 2, 7, 12, ...
Solution:
d = -3 - (-8) = -3 + 8 = 5
Check: 2 - (-3) = 5, 7 - 2 = 5, 12 - 7 = 5. ✓
Note: The AP starts with negative terms and transitions to positive terms. The common difference is positive because the sequence is increasing.
Answer: d = 5.
Example 10: Finding d When Three Terms of an AP are Given in Terms of a Variable
Problem: If 2k + 1, 3k + 3, and 5k + 1 are three consecutive terms of an AP, find k and d.
Solution:
For three terms to be in AP, the common difference between consecutive terms must be equal:
(3k + 3) - (2k + 1) = (5k + 1) - (3k + 3)
k + 2 = 2k - 2
k = 4
The three terms become: 2(4)+1 = 9, 3(4)+3 = 15, 5(4)+1 = 21.
d = 15 - 9 = 6
Check: 21 - 15 = 6 ✓
Answer: k = 4 and d = 6.
Real-World Applications
The common difference appears in many practical situations, making it one of the most applicable concepts in school mathematics:
Salary Increments: If an employee receives a fixed annual raise (say Rs. 2000 per year), their yearly salary forms an AP with d = 2000. The salary after n years can be predicted using the AP formula. For instance, starting at Rs. 30,000, the salaries would be 30000, 32000, 34000, ... — an AP with d = 2000.
Depreciation: Certain assets depreciate by a fixed amount each year (straight-line depreciation). The asset's value each year forms a decreasing AP with a negative common difference. A machine worth Rs. 5,00,000 depreciating by Rs. 50,000 per year gives the AP: 500000, 450000, 400000, ... with d = -50000.
Seating Arrangements: In a triangular seating arrangement, if each row has 2 more seats than the previous row, the number of seats per row forms an AP with d = 2. If the first row has 10 seats, the sequence is 10, 12, 14, 16, ... which helps planners calculate total seating capacity.
Temperature Patterns: If temperature rises by 1.5°C each hour during a sunny morning, the hourly temperatures form an AP with d = 1.5. Starting at 20°C at 7 AM, the readings would be 20, 21.5, 23, 24.5, 26 — an AP that helps meteorologists make short-term predictions.
Loan Repayment: In some repayment schemes, the monthly installment increases by a fixed amount each month, forming an AP. Starting at Rs. 5000 and increasing by Rs. 200 each month, the payments are 5000, 5200, 5400, ... with d = 200.
Sports Training: Athletes who increase their training distance by a fixed amount each day are following an AP pattern. A runner who starts at 2 km and adds 0.5 km daily has distances 2, 2.5, 3, 3.5, ... with d = 0.5.
Stacking Objects: Logs stacked in layers with each layer having one fewer log than the layer below form a decreasing AP. If the bottom layer has 15 logs, the stack might be 15, 14, 13, ... with d = -1.
Taxi and Ride Fares: Many taxi services charge a base fare plus a fixed rate per kilometre. The total fare for 1 km, 2 km, 3 km, ... forms an AP where d equals the per-kilometre charge.
Planting Trees: In afforestation drives, if each student plants 3 more trees than the previous student, the number of trees per student forms an AP with d = 3.
Key Points to Remember
- The common difference d = an+1 - an (next term minus current term). The order of subtraction matters — it is always later term minus earlier term.
- d is the same for all pairs of consecutive terms in an AP. If even one pair gives a different difference, the sequence is NOT an AP.
- d > 0: increasing AP. d < 0: decreasing AP. d = 0: constant sequence (all terms identical).
- An AP is completely determined by its first term a and common difference d. These two values generate the entire infinite sequence.
- The general term an = a + (n - 1)d shows that terms grow linearly with n. This linearity is the hallmark of an AP.
- For non-consecutive terms: d = (am - an) / (m - n). This is derived from the nth term formula applied to both terms.
- If an = pn + q (linear in n), then d = p and a = p + q. Any sequence whose nth term is a linear function of n is an AP.
- Conversely, if an is a non-linear function of n (e.g., an = n2), the sequence is NOT an AP.
- To check if a sequence is an AP, compute all consecutive differences and verify they are equal.
- If three numbers a, b, c are in AP, then b - a = c - b, i.e., 2b = a + c. The middle term b is called the arithmetic mean of a and c.
- The common difference can be any real number — positive, negative, zero, integer, fraction, decimal, or even irrational (e.g., d = √2 gives the AP: 1, 1+√2, 1+2√2, ...).
- A finite AP has a definite number of terms but uses the same common difference throughout.
Practice Problems
- Find the common difference of the AP: 7, 13, 19, 25, ...
- Find d for the AP: 1, -1, -3, -5, ...
- The 5th term of an AP is 30 and the 10th term is 55. Find d.
- Check whether 1, 4, 9, 16 is an AP.
- If 3, k, 11 are in AP, find k.
- The first term of an AP is -5 and the common difference is 3. Write the first 6 terms.
Frequently Asked Questions
Q1. What is the common difference of an AP?
The common difference (d) is the constant value obtained by subtracting any term from the next term in an arithmetic progression. It is the same throughout the sequence: d = a₂ - a₁ = a₃ - a₂ = ... = aₙ₊₁ - aₙ.
Q2. Can the common difference be negative?
Yes. A negative common difference means each term is less than the previous one, creating a decreasing AP. For example, 20, 15, 10, 5 has d = -5.
Q3. Can the common difference be zero?
Yes. If d = 0, every term equals the first term, giving a constant sequence like 7, 7, 7, 7. This is still a valid AP.
Q4. How do I find d if I know two non-consecutive terms?
Use the formula d = (aₘ - aₙ) / (m - n), where aₘ is the mth term and aₙ is the nth term. For example, if the 3rd term is 11 and the 8th term is 31, d = (31 - 11)/(8 - 3) = 20/5 = 4.
Q5. How do I check if a sequence is an AP?
Compute the difference between every pair of consecutive terms. If all differences are equal, the sequence is an AP with that common difference. If any two differences differ, it is NOT an AP.
Q6. If aₙ = 5n + 2, what is d?
When the nth term is a linear function of n (aₙ = pn + q), the common difference d equals the coefficient of n. Here d = 5. You can verify: a₁ = 7, a₂ = 12, d = 12 - 7 = 5.
Q7. What if the first term is negative and d is positive?
The AP starts negative and increases. Eventually, the terms become zero and then positive. For example, a = -10, d = 3 gives -10, -7, -4, -1, 2, 5, ...
Q8. Can the common difference be a fraction or decimal?
Yes, d can be any real number. For example, 1, 1.5, 2, 2.5 has d = 0.5 and 1/4, 3/4, 5/4, 7/4 has d = 1/2.
Q9. What is the importance of d in AP formulas?
The common difference d appears in both the nth term formula (aₙ = a + (n-1)d) and the sum formula (Sₙ = n/2[2a + (n-1)d]). Together with the first term a, d completely determines the entire AP.
Q10. If three numbers a, b, c are in AP, what is the relation between them?
If a, b, c are in AP, then b - a = c - b, which gives 2b = a + c. The middle term b is called the arithmetic mean of a and c.
