Checking if a Sequence is an AP
An arithmetic progression (AP) is a sequence where each term differs from the previous one by a constant value called the common difference. Before applying AP formulas, it is essential to verify that a given sequence is indeed an AP.
To check if a sequence is an AP, compute the difference between consecutive terms. If all consecutive differences are equal, the sequence is an AP. If even one difference is different, it is NOT an AP.
This skill is tested in CBSE Class 10 as a direct question and also as a prerequisite step before finding nth terms or sums.
What is Checking if Sequence is an AP?
Definition: A sequence a₁, a₂, a₃, a₄, ... is an arithmetic progression if and only if:
a₂ − a₁ = a₃ − a₂ = a₄ − a₃ = ... = constant (d)
Equivalently:
- For any three consecutive terms, the middle term is the arithmetic mean of the other two.
- aₖ₊₁ − aₖ = d for ALL values of k.
The common difference d can be:
- Positive: Increasing AP (e.g., 3, 7, 11, 15, ...)
- Negative: Decreasing AP (e.g., 20, 17, 14, 11, ...)
- Zero: Constant sequence (e.g., 5, 5, 5, 5, ...)
Checking if a Sequence is an AP Formula
Test for AP:
Check: a₂ − a₁ = a₃ − a₂ = a₄ − a₃ = ...
If all differences are equal → AP with common difference d = a₂ − a₁.
If any difference is unequal → NOT an AP.
Alternative test (for three numbers a, b, c):
a, b, c are in AP if and only if 2b = a + c
Derivation and Proof
Why 2b = a + c works:
- If a, b, c are in AP, then b − a = c − b (common difference).
- Rearranging: b − a = c − b
- Adding b to both sides: 2b − a = c
- Rearranging: 2b = a + c
This means: b = (a + c)/2, i.e., the middle term is the arithmetic mean of the first and third terms.
Extending to four numbers a, b, c, d in AP:
- Check: b − a = c − b = d − c
- All three differences must be equal.
- Also: a + d = b + c (sum of extremes = sum of middle terms).
Types and Properties
Types of questions on checking AP:
- Type 1: Given a list of numbers — check if they form an AP.
- Type 2: Given a formula for the nth term aₙ — check if it generates an AP.
- Type 3: Given three algebraic expressions — find the value of a variable so they form an AP.
- Type 4: Given a pattern (e.g., costs, distances) — determine if it follows an AP.
Common sequences that are NOT APs:
- Geometric progressions: 2, 6, 18, 54, ... (ratio is constant, not difference)
- Squares: 1, 4, 9, 16, ... (differences 3, 5, 7 — not constant)
- Powers of 2: 1, 2, 4, 8, ... (differences 1, 2, 4 — not constant)
Methods
Method 1: Compute consecutive differences
- List the terms in order.
- Compute a₂ − a₁, a₃ − a₂, a₄ − a₃, etc.
- If ALL differences are equal → AP. Otherwise → NOT an AP.
Method 2: Check 2b = a + c (for three terms)
- If three terms a, b, c are given, compute 2b and a + c.
- If 2b = a + c → they are in AP.
Method 3: Check nth term formula
- If aₙ is given as a formula in n, the sequence is an AP if and only if aₙ is a linear expression in n (i.e., aₙ = An + B where A and B are constants).
- If aₙ involves n², √n, or 1/n, it is NOT an AP.
Solved Examples
Example 1: Simple Number Sequence — Is AP
Problem: Check whether 3, 7, 11, 15, 19 is an AP.
Solution:
Compute differences:
- a₂ − a₁ = 7 − 3 = 4
- a₃ − a₂ = 11 − 7 = 4
- a₄ − a₃ = 15 − 11 = 4
- a₅ − a₄ = 19 − 15 = 4
All differences = 4 (constant).
Answer: Yes, it is an AP with common difference d = 4.
Example 2: Sequence That is NOT an AP
Problem: Check whether 1, 4, 9, 16, 25 is an AP.
Solution:
Compute differences:
- a₂ − a₁ = 4 − 1 = 3
- a₃ − a₂ = 9 − 4 = 5
- a₄ − a₃ = 16 − 9 = 7
- a₅ − a₄ = 25 − 16 = 9
Differences: 3, 5, 7, 9 — NOT constant.
Answer: No, this is NOT an AP. (These are perfect squares: 1², 2², 3², 4², 5².)
Example 3: Negative Common Difference
Problem: Is the sequence 50, 44, 38, 32, 26 an AP?
Solution:
Compute differences:
- 44 − 50 = −6
- 38 − 44 = −6
- 32 − 38 = −6
- 26 − 32 = −6
All differences = −6 (constant).
Answer: Yes, it is an AP with d = −6.
Example 4: Fractional Sequence
Problem: Check if 1/3, 5/6, 4/3, 11/6 is an AP.
Solution:
Convert to sixths: 2/6, 5/6, 8/6, 11/6
Compute differences:
- 5/6 − 2/6 = 3/6 = 1/2
- 8/6 − 5/6 = 3/6 = 1/2
- 11/6 − 8/6 = 3/6 = 1/2
All differences = 1/2 (constant).
Answer: Yes, it is an AP with d = 1/2.
Example 5: Using 2b = a + c to Find Unknown
Problem: Find the value of k such that k, 2k + 1, 3k + 5 are in AP.
Solution:
Given: a = k, b = 2k + 1, c = 3k + 5
For AP: 2b = a + c
- 2(2k + 1) = k + (3k + 5)
- 4k + 2 = 4k + 5
- 2 = 5
This is a contradiction — no value of k satisfies this.
Answer: No value of k makes these three expressions an AP.
Example 6: Finding k for AP
Problem: Find k if k + 1, 3k, 4k + 2 are in AP.
Solution:
Given: a = k + 1, b = 3k, c = 4k + 2
For AP: 2b = a + c
- 2(3k) = (k + 1) + (4k + 2)
- 6k = 5k + 3
- k = 3
Verify: k = 3 → terms are 4, 9, 14. Differences: 5, 5. ✓ AP.
Answer: k = 3.
Example 7: Checking an nth Term Formula
Problem: The nth term of a sequence is aₙ = 3n + 5. Is it an AP? If so, find d.
Solution:
Method 1 — Check if aₙ is linear in n:
- aₙ = 3n + 5 is of the form An + B (linear).
- Therefore, it IS an AP with d = coefficient of n = 3.
Method 2 — Verify by computing terms:
- a₁ = 3(1) + 5 = 8
- a₂ = 3(2) + 5 = 11
- a₃ = 3(3) + 5 = 14
- Differences: 11 − 8 = 3, 14 − 11 = 3. ✓
Answer: Yes, it is an AP with d = 3, first term a = 8.
Example 8: nth Term Formula That is NOT an AP
Problem: Is the sequence given by aₙ = n² + 1 an AP?
Solution:
Check: aₙ = n² + 1 involves n² → NOT linear in n → NOT an AP.
Verify:
- a₁ = 1 + 1 = 2
- a₂ = 4 + 1 = 5
- a₃ = 9 + 1 = 10
- a₄ = 16 + 1 = 17
- Differences: 3, 5, 7 — NOT constant.
Answer: No, this sequence is NOT an AP.
Example 9: Constant Sequence
Problem: Is the sequence 7, 7, 7, 7, ... an AP?
Solution:
Compute differences:
- 7 − 7 = 0
- 7 − 7 = 0
- 7 − 7 = 0
All differences = 0 (constant).
Answer: Yes, it is an AP with d = 0.
Example 10: Real-Life AP Check
Problem: A taxi charges ₹15 for the first km, ₹23 for 2 km, ₹31 for 3 km, ₹39 for 4 km. Do the charges form an AP?
Solution:
Charges: 15, 23, 31, 39
Differences:
- 23 − 15 = 8
- 31 − 23 = 8
- 39 − 31 = 8
All differences = 8 (constant).
Answer: Yes, the charges form an AP with first term a = 15 and common difference d = 8.
Real-World Applications
Why checking for AP matters:
- Before using AP formulas: The nth term formula aₙ = a + (n−1)d and sum formula Sₙ = n/2[2a + (n−1)d] only work if the sequence is actually an AP.
- Pattern recognition: Identifying APs in data helps predict future terms.
- Financial planning: Fixed monthly savings, equal EMI increments, and salary hikes at constant amounts form APs.
- Physics: Uniformly accelerated motion produces positions at equal time intervals that form an AP.
- Exam problems: CBSE often asks "Which of these sequences is an AP?" as a 1–2 mark question.
Key Points to Remember
- A sequence is an AP if the difference between consecutive terms is constant.
- Common difference d = a₂ − a₁. Check that ALL consecutive differences equal d.
- The common difference can be positive, negative, or zero.
- For three terms a, b, c: they are in AP if and only if 2b = a + c.
- If aₙ = An + B (linear in n) → AP with d = A. If aₙ involves n², √n, 1/n → NOT an AP.
- A constant sequence (e.g., 5, 5, 5, ...) IS an AP with d = 0.
- Squares (1, 4, 9, 16, ...) and cubes (1, 8, 27, ...) are NOT APs.
- Checking even ONE unequal difference is enough to conclude it is NOT an AP.
- Always compute at least 3 consecutive differences to be safe.
- This is a prerequisite skill — verify AP before applying any AP formula.
Practice Problems
- Check if the following sequences are APs: (i) 2, 4, 8, 16 (ii) −3, −1, 1, 3, 5 (iii) 1, 1, 2, 3, 5, 8
- The nth term of a sequence is aₙ = 2n − 7. Is it an AP? If yes, find the first term and common difference.
- Find x if x − 1, x + 3, 3x − 1 are in AP.
- Is the sequence √2, √8, √18, √32, ... an AP? (Hint: simplify each term first.)
- Check whether the sequence given by aₙ = 3 − 4n is an AP. Find the 10th term.
- The cost of digging a well is ₹200 for the first metre, ₹200 for the second, ₹250 for the third, ₹250 for the fourth. Is this an AP?
Frequently Asked Questions
Q1. How do you check if a sequence is an AP?
Compute the differences between consecutive terms. If all differences are equal, it is an AP. If any difference is different from the others, it is NOT an AP.
Q2. Can the common difference be zero?
Yes. A constant sequence like 5, 5, 5, 5 is an AP with d = 0.
Q3. Is 1, 4, 9, 16, 25 an AP?
No. The differences are 3, 5, 7, 9 — not constant. These are perfect squares, not an AP.
Q4. How do you check if an nth term formula gives an AP?
If aₙ is a linear function of n (form An + B), the sequence is an AP with d = A. If aₙ involves n², n³, √n, or 1/n, it is NOT an AP.
Q5. What does 2b = a + c mean?
Three numbers a, b, c are in AP if and only if 2b = a + c. This means the middle term b is the arithmetic mean of a and c.
Q6. Is 2, 4, 8, 16 an AP?
No. The differences are 2, 4, 8 — not constant. This is a geometric progression (GP) with common ratio 2, not an AP.
Q7. Can negative numbers form an AP?
Yes. For example, −10, −6, −2, 2, 6 is an AP with d = 4. The common difference can also be negative: 10, 7, 4, 1, −2 has d = −3.
Q8. Is it enough to check just two differences?
For a short sequence of 3 terms, yes. For longer sequences, check all consecutive differences. In exam problems, typically 4–5 terms are given, so check all differences.
