ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Sequences and Series (Set 2): 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 arithmetic mean of two positive numbers is 5 and their geometric mean is 4. The numbers are:
Answer: D. 8 and 2
Explanation: a + b = 10 and ab = 16, so a and b are the roots of t² − 10t + 16 = 0, i.e. 8 and 2.
Q2. In an A.P., the 5th term is 17 and the 12th term is 45. The 20th term is:
Answer: A. 77
Explanation: 7d = 45 − 17 = 28 ⇒ d = 4, a = 17 − 16 = 1, so a₂₀ = 1 + 19·4 = 77.
Q3. Two geometric means are inserted between 3 and 81. They are:
Answer: D. 9 and 27
Explanation: 3, 3r, 3r², 3r³ with 3r³ = 81 ⇒ r = 3, so the means are 9 and 27.
Q4. A sequence whose reciprocals form an arithmetic progression is called:
Answer: B. a harmonic progression
Explanation: By definition a, b, c, … is an H.P. when 1/a, 1/b, 1/c, … form an A.P.
Q5. If the sum of the first n terms of a sequence is Sₙ = 2ⁿ − 1, then its nth term is:
Answer: B. 2ⁿ⁻¹
Explanation: aₙ = Sₙ − Sₙ₋₁ = (2ⁿ − 1) − (2ⁿ⁻¹ − 1) = 2ⁿ⁻¹ (also a₁ = S₁ = 1 fits).
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Q6. The sum of the infinite series 1 + 3(1/3) + 5(1/3)² + 7(1/3)³ + … is:
Q7. The sum of the first n natural numbers is 210. The value of n is:
Q8. In an A.P., the first term is 2 and the sum of the first 10 terms is 155. The common difference is:
Q9. The sum of all two-digit natural numbers divisible by 3 is:
Q10. The sum of the squares of the first ten odd numbers, 1² + 3² + 5² + … + 19², is:
Q11. The common difference of the A.P. whose nth term is aₙ = 5 − 2n is:
Q12. If the nth term of a sequence is aₙ = n² − n + 1, then a₅ is:
Q13. If a, b, c are in harmonic progression, then b equals:
Q14. The 6th term of the G.P. 2, −6, 18, … is:
Q15. Three numbers in G.P. have product 216 and sum 19. The largest of them is:
Q16. The sum of the first 20 terms of the A.P. 1 + 4 + 7 + … is:
Q17. How many terms of the A.P. 3 + 5 + 7 + … must be taken to make the sum 120?
Q18. The value of Σ (2k + 1) for k = 1 to 10 is:
Q19. The recurring decimal 0.2333… (only the 3 repeating) is equal to:
Q20. If a, b, c (all positive) are in G.P., then log a, log b, log c are in:
Q21. Five arithmetic means are inserted between 4 and 16. Their sum is:
Q22. The sum of the infinite geometric series 6 − 3 + 3/2 − 3/4 + … is:
Q23. The sum of the squares of the terms of the infinite G.P. 1 + 1/2 + 1/4 + … , i.e. 1 + 1/4 + 1/16 + …, is:
Q24. In an A.P., the pth term is q and the qth term is p (p ≠ q). The (p + q)th term is:
Q25. A sequence is defined by a₁ = 1, a₂ = 1 and aₙ = aₙ₋₁ + aₙ₋₂ for n ≥ 3. The value of a₈ is:
Q26. How many terms are there in the G.P. 2, 4, 8, …, 1024?
Q27. The 15th term of the arithmetic progression 10, 7, 4, … is:
Q28. The sum of the first 6 terms of the G.P. 1 + 3 + 9 + … is:
Q29. For two distinct positive numbers with arithmetic mean A, geometric mean G and harmonic mean H, the correct relation is:
Q30. The harmonic mean between 3 and 6 is: