20 Solved AP GP Formula Questions for CAT 2026 Practice
Twenty AP GP formula questions, four clusters of five, with step-by-step solutions calibrated to CAT difficulty. Cluster 1 covers AP basics (nth term and sum). Cluster 2 covers GP fundamentals through infinite GP. Cluster 3 covers AGP and special-series problems. Cluster 4 covers HP and the AM-GM-HM family.
Each question is paired with a 3-step solution and a final answer. Run the set untimed first for accuracy, then timed for speed, then mixed-order for recognition. The set works as a calibration anchor: re-run it three times across the prep cycle to track progress on AP GP formula questions before CAT 2026.
- Cluster 1 (Q1-Q5): AP nth term, AP sum forms 1 and 2, AM, sum of natural numbers, sum of multiples.
- Cluster 2 (Q6-Q10): GP nth term, finite GP sum, infinite GP convergence, GM, repeating decimal disguise.
- Cluster 3 (Q11-Q15): AGP general term, sum of squares, sum of cubes, sum of odd squares, mixed AGP-AP.
- Cluster 4 (Q16-Q20): HP nth term via AP, harmonic mean for equal-distance speed, AM-GM-HM inequality, AM-HM-GM identity, mean-relationship problem.
Why Step-by-Step AP GP Practice Beats Random Drills
Random-order practice trains formula recall. Clustered, step-by-step practice trains pattern recognition, which is the actual bottleneck in CAT QA. Aspirants who run 100 random AP GP questions and check only the final answers improve their accuracy slowly because the recognition step is invisible. Aspirants who run 30 questions in clustered batches with step-tagged review improve faster because every miss carries diagnostic information.
The 20-question set below is structured for clustered review. The full 24-formula AP GP cheatsheet is the reference; the 6 AP GP formula shortcuts guide compresses solve time; the AP GP vs HP comparison sets the recurrence priority across the three progressions. Use all three alongside this practice set, and consult the CAT quantitative aptitude syllabus for how progressions sit inside the broader QA topic mix.
Find the 20th term of the AP 5, 12, 19, 26, ...
Step 1: Identify a = 5, d = 7. Step 2: Apply an = a + (n − 1)d = 5 + 19 × 7. Step 3: 5 + 133 = 138.
Answer: 138
Find the sum of the first 25 terms of an AP with first term 3 and common difference 4.
Step 1: a = 3, d = 4, n = 25. Step 2: Apply S = (n / 2)[2a + (n − 1)d] = (25 / 2)[6 + 96]. Step 3: 12.5 × 102 = 1275.
Answer: 1275
Find the sum of all multiples of 9 between 1 and 200.
Step 1: First multiple a = 9, last multiple l = 198. Step 2: n = (198 − 9)/9 + 1 = 22. Step 3: S = (22 / 2)(9 + 198) = 11 × 207 = 2277.
Answer: 2277
Insert 3 arithmetic means between 5 and 25.
Step 1: 3 means inserted means 5 terms total: 5, m₁, m₂, m₃, 25. Step 2: Common difference d = (25 − 5) / (3 + 1) = 5. Step 3: Means are 10, 15, 20.
Answer: 10, 15, 20
Find the sum of natural numbers from 50 to 100 inclusive.
Step 1: Sum1 to 100 = 100 × 101 / 2 = 5050. Step 2: Sum1 to 49 = 49 × 50 / 2 = 1225. Step 3: 5050 − 1225 = 3825.
Answer: 3825
In a GP, first term is 3, common ratio is 2. Find the 8th term.
Step 1: a = 3, r = 2, n = 8. Step 2: an = a r(n − 1) = 3 × 27. Step 3: 3 × 128 = 384.
Answer: 384
Find the sum of the first 6 terms of the GP 2, 6, 18, 54, ...
Step 1: a = 2, r = 3, n = 6. Step 2: S = a(rn − 1)/(r − 1) = 2(36 − 1)/(3 − 1). Step 3: 2 × 728 / 2 = 728.
Answer: 728
Find the infinite sum of 4, 2, 1, 1/2, ...
Step 1: a = 4, r = 1/2. Step 2: Verify |r| < 1 ✓. Step 3: S = a / (1 − r) = 4 / (1 − 0.5) = 8.
Answer: 8
Find the geometric mean of 4 and 25.
Step 1: GM formula √(ab). Step 2: √(4 × 25) = √100. Step 3: 10.
Answer: 10
Convert 0.454545... to a fraction.
Step 1: Tag as infinite GP with a = 0.45, r = 0.01. Step 2: Verify |r| < 1 ✓. Step 3: S = 0.45 / 0.99 = 45 / 99 = 5 / 11.
Answer: 5 / 11
Halfway through the practice set. Want a CAT 2026 plan that schedules these 4 clusters across your weekly QA cycle with mock-tagged tracking?
Schedule My AP GP PracticeFind the 5th term of the AGP 2, 6, 18, 54, ... wait, this is GP. Adjust: find the 5th term of 1, 2·3, 3·9, 4·27, 5·81 where the AP component is 1, 2, 3, 4 and GP ratio is 3.
Step 1: AGP general term [a + (n − 1)d] r(n − 1) with a = 1, d = 1, r = 3. Step 2: 5th term: [1 + 4 × 1] × 34 = 5 × 81. Step 3: 405.
Answer: 405
Find the sum of the first 10 perfect squares.
Step 1: n = 10. Step 2: Apply n(n + 1)(2n + 1) / 6 = 10 × 11 × 21 / 6. Step 3: 2310 / 6 = 385.
Answer: 385
Find 1³ + 2³ + 3³ + ... + 8³.
Step 1: n = 8. Step 2: Apply [n(n + 1) / 2]² = [8 × 9 / 2]² = 36². Step 3: 1296.
Answer: 1296
Find the sum of the squares of the first 6 odd natural numbers.
Step 1: Odd squares formula n(2n − 1)(2n + 1) / 3 with n = 6. Step 2: 6 × 11 × 13 / 3 = 858 / 3. Step 3: 286. Verify: 1 + 9 + 25 + 49 + 81 + 121 = 286 ✓.
Answer: 286
Find the sum of the first 5 terms of the AGP 1, 4, 12, 32, 80 where AP component is 1, 2, 3, 4, 5 and GP ratio is 2.
Step 1: Use subtract-and-shift. S = 1 + 4 + 12 + 32 + 80 = 129 by direct sum since n is small. Step 2: Verify general term: [1 + (n − 1) × 1] × 2(n − 1). Step 3: Direct addition gives 129.
Answer: 129
Find the 6th term of the HP 1/2, 1/5, 1/8, 1/11, ...
Step 1: Reciprocals form an AP: 2, 5, 8, 11, ... with a = 2, d = 3. Step 2: 6th term of AP = 2 + 5 × 3 = 17. Step 3: HP 6th term = 1 / 17.
Answer: 1 / 17
A train covers 200 km at 50 km/h and another 200 km at 75 km/h. Find average speed for the full trip.
Step 1: Equal distances at two speeds: HM applies. Step 2: HM = 2 × 50 × 75 / (50 + 75) = 7500 / 125. Step 3: 60 km/h.
Answer: 60 km/h
For positive reals x and y with x + y = 16, find the maximum of xy.
Step 1: Tag as AM-GM equality lock. Step 2: AM = 8 ≥ √(xy) = GM; equality at x = y. Step 3: Max xy = 8 × 8 = 64.
Answer: 64
For two positive numbers, AM = 25 and HM = 16. Find GM.
Step 1: Apply GM² = AM · HM. Step 2: GM² = 25 × 16 = 400. Step 3: GM = 20.
Answer: 20
Find the harmonic mean of 4 and 12.
Step 1: HM = 2ab / (a + b). Step 2: HM = 2 × 4 × 12 / (4 + 12) = 96 / 16. Step 3: 6.
Answer: 6
How to Use This 20-Question Practice Set
The set works best as a calibration anchor across three runs. Each run targets a different skill: accuracy, speed, recognition. Aspirants who run the set three times during prep see consistent percentile lift on AP GP CAT questions.
- Run 1, untimed accuracy. Solve all 20 without a clock. Check answer plus solution path. Tag every miss against a specific formula on the 24-formula AP GP cheatsheet.
- Run 2, timed speed. Solve in clusters of 5 with a 7 to 10 minute cap per cluster. Compare time-per-question against the 60 to 90 second target. The 6 AP GP formula shortcuts guide compresses the slow ones.
- Run 3, mixed-order recognition. Shuffle all 20. Solve in random order. The drill trains recognition speed, the bottleneck in CAT QA. Tag every error against the AP GP pattern map.
The mock analysis framework covers how to track these three runs against full-length mocks, while the 99 percentile playbook covers how AP GP fits inside the broader QA mix.
Three Errors That Recur Across the Practice Set
Across cohorts running this 20-question set, three errors keep showing up. Each is preventable; each is the difference between 90-percentile and 99-percentile accuracy on AP GP.
Applying a / (1 − r) on Q8 without first verifying |r| < 1. The condition holds in this set, but a CAT variant could give r = 1.2 and the same reflex would produce a wrong answer. Always tag the condition.
Computing (50 + 75) / 2 = 62.5 instead of HM = 60. The reflex AM is 2.5 km/h off the correct answer; in CAT, that 2.5 mismatch makes options indistinguishable and forces a guess. Lock HM for any equal-distance speed.
Using 28 instead of 27 for the 8th term. The exponent is n − 1, not n. CAT regularly tests this exact off-by-one in disguise. Verify by writing out the first 3 to 4 terms before applying the formula.
Re-run the set monthly across the prep cycle. Accuracy should climb in run 1; solve time should drop in run 2; mixed-order miss-rate should fall in run 3. If any of the three metrics plateaus, the leak is in that specific dimension. Tagged retrospective review (with the CAT score predictor as the percentile-translation tool) compounds across mocks.
Common Doubts on AP GP Practice for CAT Answered
Practice in clusters by progression family. Start with 5 AP basics, then 5 GP, then 5 AGP/special series, then 5 HP/means. Time each cluster: 10 minutes for the first, 7 for the second, 5 for the next two. Clustered structure builds recognition speed.
60 to 90 seconds per question on average: 10s identify family, 10s lock formula, 30s apply, 20s verify, 10-20s buffer for harder problems. Beginners can start at 120s and compress as recognition improves.
AP nth term and sum first, GP nth term and finite sum next, infinite GP and convergence third, AM-GM-HM and means fourth, AGP and special series fifth, HP via harmonic mean last. Order matches CAT recurrence and dependency.
About 80 percent yes, with the six high-recurrence shortcuts. The remaining 20 percent require AGP subtract-and-shift or HP-to-AP conversion. Aspirants who only practice shortcuts cap at 90-95 percentile; those who add the standard algebra clear 99-plus.
Step-by-step solutions reveal the recognition step, formula choice, substitution, and verification in sequence. Aspirants who only check final answers lose information about whether their tag was correct. Step-tagged review converts misses into corrective lessons.
100 to 150 questions across the full prep cycle is enough for tier-1 percentile coverage. 4 sessions per week of 5 to 8 questions across 6 to 8 weeks. Quality of review matters more than question volume.
The Practice-Set Recap
- Discipline 1Practice in clusters by progression family, not in random order.
- Discipline 2Run the set three times: untimed accuracy, timed speed, mixed-order recognition.
- Discipline 3Tag every miss against a specific formula on the master cheatsheet.
- Discipline 4Verify the convergence condition |r| < 1 before applying infinite GP.
- Discipline 5Step-by-step review beats answer-key-only review on percentile lift.
- Discipline 6100 to 150 well-tagged questions outperform 300 quickly solved ones.
Plan AP GP practice across the full CAT 2026 prep cycle
A CAT 2026 plan that schedules these 4 clusters across your weekly QA cycle, tracks step-tagged accuracy across mocks, and runs the calibration set three times before exam day.
Schedule My AP GP Practice
