ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Sequences and Series (Set 7): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer review. Covers…
What this test covers
The first 5 questions of this test, with the correct answer and explanation.
Q1. The value of 1·2·3 + 2·3·4 + 3·4·5 + 4·5·6 + 5·6·7 + 6·7·8 is
Answer: B. 756
Explanation: Σ k(k + 1)(k + 2) from 1 to n equals n(n + 1)(n + 2)(n + 3)/4. For n = 6: 6·7·8·9/4 = 756.
Q2. The sum 2² + 4² + 6² + … + 20² equals
Answer: D. 1540
Explanation: Σ(2k)² = 4Σk² = 4·(10·11·21/6) = 4·385 = 1540.
Q3. The sum 11³ + 12³ + 13³ + … + 20³ equals
Answer: D. 41075
Explanation: Σ₁²⁰ k³ = (210)² = 44100 and Σ₁¹⁰ k³ = 55² = 3025. The difference is 41075.
Q4. The sums of the first m and first n terms of an A.P. are in the ratio m² : n². The ratio of its 9th term to its 5th term is
Answer: B. 17 : 9
Explanation: Sₙ = cn² gives aₖ = c(k² − (k − 1)²) = c(2k − 1). So a₉ : a₅ = 17 : 9. (The squares ratio 81 : 25 is the ratio of the sums, not of the terms.)
Q5. In an A.P. the 3rd term is 1/4 and the 4th term is 1/3. The sum of its first 12 terms is
Answer: B. 13/2
Explanation: d = 1/3 − 1/4 = 1/12 and a₁ = 1/4 − 2/12 = 1/12. S₁₂ = 12/2·(2·(1/12) + 11·(1/12)) = 6·(13/12) = 13/2.
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Q6. The sum of the first n terms of a sequence is Sₙ = 3ⁿ − 1. Its 4th term is
Q7. The sum of the reciprocals of the first five terms of the G.P. 3, 6, 12, 24, 48 is
Q8. For x > 0, the least value of 2x + 18/x is
Q9. The nth term of a series is 2n² − 1. The sum of its first 10 terms is
Q10. A sequence has a₁ = 3 and aₙ₊₁ = aₙ + 2n. Then a₁₀ equals
Q11. A sequence has a₁ = 1 and aₙ₊₁ = aₙ/(1 + aₙ). Then a₂₀ equals
Q12. A sequence has a₁ = 5, a₂ = 8 and aₙ₊₂ = 2aₙ₊₁ − aₙ. Then a₁₀ equals
Q13. Logs are stacked in rows, with 15 logs in the bottom row and one log fewer in each higher row, up to a top row of 5 logs. The total number of logs is
Q14. The sum of all three-digit natural numbers that are divisible by 7 is
Q15. The sum of all integers from 1 to 100 that are divisible by 4 or by 6 is
Q16. m arithmetic means are inserted between 1 and 31. If the 7th mean and the (m − 1)th mean are in the ratio 5 : 9, then m equals
Q17. Three geometric means are inserted between 1 and 256, with all terms positive. Their sum is
Q18. The sum to infinity of 1 − 2/3 + 3/9 − 4/27 + … is
Q19. The four angles of a quadrilateral are in A.P., and the largest is four times the smallest. The second largest angle is
Q20. A square of side 8 cm is given. A second square is formed by joining the midpoints of its sides, a third by joining midpoints of the second, and so on indefinitely. The total area of all the squares is
Q21. The sequence aₙ = sin(nπ/2) takes the values 1, 0, −1, 0, 1, 0, … The sum of its first 2026 terms is
Q22. The sum 1/(1·4) + 1/(4·7) + 1/(7·10) + … + 1/(28·31) equals
Q23. Which of the following sequences is both an arithmetic progression and a geometric progression?
Q24. In an A.P., the sum of the 5th and 26th terms is 90. The sum of the first 30 terms is
Q25. In an infinite G.P. with positive terms, every term equals the sum of all the terms that follow it. The common ratio is
Q26. The least number of terms of 1 + 3 + 5 + 7 + … needed for the sum to exceed 2000 is
Q27. Three positive numbers are in A.P. with sum 15. Adding 1, 4 and 19 respectively gives three numbers in G.P. The largest of the original numbers is
Q28. A bacterial culture starts with 3000 cells and doubles every 5 hours. The number of cells after 20 hours is
Q29. A car goes uphill from A to B at 40 km/h and returns downhill along the same road at 60 km/h. Its average speed for the round trip is
Q30. Two positive numbers have arithmetic mean : geometric mean = 5 : 4. The ratio of the larger number to the smaller is