ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Sequences and Series (Set 4): 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. In an arithmetic progression the 5th term is 19 and the 12th term is 47. The 20th term is:
Answer: B. 79
Explanation: a + 4d = 19 and a + 11d = 47 give 7d = 28, d = 4 and a = 3. Then a₂₀ = 3 + 19 × 4 = 79.
Q2. The sum of the first n terms of a sequence is Sₙ = 4n² − n. Its 10th term is:
Answer: C. 75
Explanation: a₁₀ = S₁₀ − S₉ = 390 − 315 = 75. (390 is the sum of ten terms, not the tenth term.)
Q3. How many terms of the A.P. 3, 7, 11, … must be taken so that their sum is 465?
Answer: D. 15
Explanation: Sₙ = (n/2)(6 + 4(n − 1)) = n(2n + 1) = 465, so 2n² + n − 465 = 0 giving n = 15 (the other root is negative).
Q4. Four arithmetic means are inserted between 7 and 27. The sum of these four means is:
Answer: D. 68
Explanation: The common difference is (27 − 7)/5 = 4, so the means are 11, 15, 19, 23 with sum 68.
Q5. In a geometric progression the 3rd term is 12 and the 6th term is 96. The 8th term is:
Answer: A. 384
Explanation: r³ = 96/12 = 8, so r = 2 and a = 12/4 = 3. Then a₈ = 3 × 2⁷ = 384.
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Q6. The sum of the first seven terms of the G.P. 5, 10, 20, … is:
Q7. An infinite G.P. has first term 8 and sum 12. Its second term is:
Q8. The recurring decimal 0.272727… written as an infinite geometric series equals the fraction:
Q9. Two geometric means are inserted between 3 and 81. Their sum is:
Q10. The 3rd term of a harmonic progression is 1/5 and its 7th term is 1/13. Its 10th term is:
Q11. The arithmetic mean of two positive numbers is 10 and their geometric mean is 8. Their harmonic mean is:
Q12. The value of 1² + 2² + 3² + … + 12² is:
Q13. The sum of the squares of the first 10 odd natural numbers, 1² + 3² + 5² + … + 19², is:
Q14. Find the value of 1·3 + 2·4 + 3·5 + … + 8·10.
Q15. For |x| < 1 the sum of the series 1 + 2x + 3x² + 4x³ + … at x = 1/3 is:
Q16. Three numbers in A.P. have sum 24 and product 440. The largest of them is:
Q17. Three positive numbers in G.P. have sum 42 and product 512. The largest of them is:
Q18. The sum 1/(1·2) + 1/(2·3) + 1/(3·4) + … + 1/(19·20) equals:
Q19. A sequence is defined by a₁ = 1 and aₙ₊₁ = 2aₙ + 3. The value of a₅ is:
Q20. In an A.P., the sum of the first 7 terms equals the sum of the first 12 terms. The sum of the first 19 terms is:
Q21. The sum of all multiples of 7 lying between 100 and 500 is:
Q22. The sum 1 + 3 + 9 + … + 3⁹ is:
Q23. The value of 1² − 2² + 3² − 4² + … + 19² − 20² is:
Q24. The numbers x, x + 2 and x + 8 are consecutive terms of a G.P. The common ratio is:
Q25. If 1/(b + c), 1/(c + a), 1/(a + b) are in A.P., which of the following is also in A.P.?
Q26. A sequence has terms 2, 6, 12, 20, 30, … The 12th term is:
Q27. The number of terms in the A.P. 5, 8, 11, …, 80 is:
Q28. The arithmetic mean of two positive numbers is 25 and their geometric mean is 15. The difference between the numbers is:
Q29. If a, b, c are in H.P. with a = 6, b = 4, c = 3, the ratio (a − b)/(b − c) equals:
Q30. In an A.P. the sum of the 4th and 8th terms is 26. The sum of the first 11 terms is: