How to Solve Arithmetic Progression Problems
Solve arithmetic progression aptitude problems using the nth-term and sum formulas, with a worked example and practice questions with answers.
Expected Interview Answer
An arithmetic progression (AP) is a sequence where each term is the previous term plus a fixed common difference d, so the nth term is a + (n−1)d and the sum of n terms is Sn = n/2 × (2a + (n−1)d).
Because the difference between consecutive terms is constant, any AP is fully determined by just two numbers: the first term a and the common difference d. The sum formula can also be written as n/2 × (first term + last term), which is often faster when the last term is already known. Three consecutive AP terms satisfy the property that the middle term equals the average of its two neighbors. Always confirm the difference is genuinely constant across the given terms before applying any AP formula, since near-constant sequences are a common distractor.
- Two numbers (a and d) fully determine an entire AP
- The n/2 × (first + last) form of the sum is often quicker to apply
- The middle-equals-average property verifies three-term APs instantly
AI Mentor Explanation
A bowler adding exactly 2 runs conceded per over as fatigue sets in — 4, 6, 8, 10 — is an arithmetic progression with common difference 2, unlike a geometric progression that would multiply instead of add. Finding the runs conceded in the 8th over is simply a + 7d, the nth-term formula applied directly. The total runs conceded across a full 10-over spell sums via n/2 × (first over’s runs + 10th over’s runs), the AP sum shortcut.
Worked example
First term & difference
- a = 4, d = 3
10th term
- T10 = 4 + 9×3 = 31
Sum of 10 terms
- S10 = 10/2 × (4+31) = 175
Step-by-Step Explanation
Step 1
Identify a and d
Subtract consecutive terms to confirm a constant common difference d.
Step 2
Apply the nth-term formula
Tn = a + (n−1)d gives any specific term.
Step 3
Apply the sum formula
Sn = n/2 × (2a + (n−1)d), or n/2 × (first + last) when the last term is known.
Step 4
Verify with the average property
For 3 consecutive terms, the middle equals the average of its neighbors.
What Interviewer Expects
- Correct identification of a and d from given terms
- Correct nth-term and sum formulas, including the first+last shortcut
- Distinguishing AP (additive) growth from GP (multiplicative) growth
- Verifying d is constant before applying any AP formula
Common Mistakes
- Using the geometric progression sum formula on an arithmetic sequence
- Off-by-one errors in (n−1)d when counting terms
- Forgetting the alternate sum form n/2 × (first + last) when it would be faster
- Sign errors when d is negative, producing a decreasing sequence
Best Answer (HR Friendly)
“An arithmetic progression adds a fixed amount d each step rather than multiplying. Once you know the first term and d, the nth term is a plus n minus 1 times d, and the sum of n terms is n over 2 times the first term plus the last term. That n over 2 times first-plus-last version is often the fastest way to compute a sum once you already know both endpoints.”
Follow-up Questions
- How do you find the number of terms in an AP given the first, last term and common difference?
- How would you insert arithmetic means between two numbers?
- How does a negative common difference change the sequence behavior?
- How do you find the sum of an AP when only the first and last terms are known, not d?
MCQ Practice
1. An AP has first term 7 and common difference 4. What is the 6th term?
T6 = 7 + 5×4 = 7 + 20 = 27.
2. Find the sum of the first 15 terms of the AP 3, 7, 11, ...
a=3, d=4. S15 = 15/2 × (2×3 + 14×4) = 7.5 × (6+56) = 7.5×62 = 465.
3. Three numbers x, 10, y are in AP. If x = 6, what is y?
Middle term equals average of neighbors: 10 = (6+y)/2 → 20 = 6+y → y = 14.
Flash Cards
AP nth-term formula? — Tn = a + (n−1)d.
AP sum formula? — Sn = n/2 × (2a + (n−1)d) = n/2 × (first term + last term).
Property of 3 consecutive AP terms? — Middle term = average of the two neighbors.
How to find common difference? — Subtract any term from the term immediately after it.