ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Sequences and Series (Set 6): 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 sum of the first n terms of an arithmetic progression is Sₙ = 3n² + 2n. The 10th term is:
Answer: D. 59
Explanation: a₁₀ = S₁₀ − S₉ = (300 + 20) − (243 + 18) = 320 − 261 = 59.
Q2. How many terms below 200 are common to the two arithmetic progressions 3, 7, 11, 15, … and 2, 9, 16, 23, …?
Answer: D. 7
Explanation: The first common term is 23, and later ones differ by lcm(4, 7) = 28: 23, 51, 79, 107, 135, 163, 191. The next, 219, exceeds 200, so there are 7.
Q3. Three positive numbers in G.P. have product 1000 and the sum of their squares is 525. The largest of them is:
Answer: D. 20
Explanation: The middle term is ∛1000 = 10, so the terms are 10/r, 10, 10r with 100(r² + 1 + 1/r²) = 525. Then r² + 1/r² = 17/4, giving r² = 4 (or ¼). The terms are 5, 10, 20, and the largest is 20.
Q4. An infinite G.P. has sum 6 and the sum of the squares of its terms is 12. Its first term is:
Answer: A. 3
Explanation: With a/(1 − r) = 6 and a²/(1 − r²) = 12: 36(1 − r)²/(1 − r²) = 12 gives 3(1 − r) = 1 + r, so r = ½ and a = 6(1 − ½) = 3.
Q5. If three harmonic means are inserted between 1/2 and 1/10, the middle one is:
Answer: B. 1/6
Explanation: The reciprocals form a 5-term A.P. from 2 to 10 with d = 2: 2, 4, 6, 8, 10. The middle harmonic mean is the reciprocal of 6, i.e. 1/6.
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Q6. Two positive numbers have arithmetic mean 10 and geometric mean 8. Their harmonic mean is:
Q7. If four arithmetic means are inserted between 3 and 28, their sum is:
Q8. The product of the three geometric means inserted between 1 and 81 (taking positive ratio) is:
Q9. The sum 1·2 + 2·3 + 3·4 + … + 12·13 equals:
Q10. The sum of the cubes of the first ten odd natural numbers, 1³ + 3³ + 5³ + … + 19³, is:
Q11. The sum to infinity of 1/4 + 2/4² + 3/4³ + 4/4⁴ + … is:
Q12. A sequence has a₁ = 1 and aₙ₊₁ = 2aₙ + n for n ≥ 1. The value of a₅ is:
Q13. A sequence has a₁ = 1, a₂ = 3 and aₙ = aₙ₋₁ + aₙ₋₂ for n ≥ 3. Then a₇ is:
Q14. If a, b, c are in G.P. (positive terms), then a + b, 2b, b + c are in:
Q15. Four numbers in A.P. have sum 20 and the sum of their squares is 120. The largest of them is:
Q16. A person saves 100 rupees in the first month and 20 rupees more each following month than in the previous month. The total saved in 24 months is:
Q17. A ball dropped from a height of 20 m rebounds to 3/5 of its previous height after each bounce. The total distance travelled before it comes to rest is:
Q18. The sum 1 + 3 + 7 + 15 + … to 10 terms (the kth term being 2ᵏ − 1) is:
Q19. The sum of the first 50 odd natural numbers minus the sum of the first 50 even natural numbers is:
Q20. The sum 1/(1 + √2) + 1/(√2 + √3) + … + 1/(√80 + √81) equals:
Q21. In a G.P. the 4th term is 3. The product of the first seven terms is:
Q22. In an A.P. the sum of the first 8 terms equals the sum of the first 12 terms. The sum of the first 20 terms is:
Q23. In an A.P. the 5th term is 12 and the 12th term is 5. The 17th term is:
Q24. The sum to infinity of Σ (2ⁿ + 3ⁿ)/6ⁿ for n = 1, 2, 3, … is:
Q25. The sum to infinity of 1/(1·2·3) + 1/(2·3·4) + 1/(3·4·5) + … is:
Q26. The first term of the G.P. 2, 6, 18, … that exceeds 10 000 is the term number:
Q27. The value of 1² − 2² + 3² − 4² + … − 20² is:
Q28. The arithmetic mean of two positive numbers is 25 and their harmonic mean is 9. Their geometric mean is:
Q29. If x, 2x + 2 and 3x + 3 are the first three terms of a G.P. (x ≠ −1), the fourth term is:
Q30. The sums of the first n terms of two A.P.s are in the ratio (3n + 1) : (2n + 5). The ratio of their 7th terms is: