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Question 1. Which of the following best describes a finite sequence? A) It has a limit as n → ∞. B) It contains an infinite number of terms. C) It has a last term after a certain index. D) Its terms are defined by a recurrence relation. Answer: C Explanation: A finite sequence stops after a specific index, so it possesses a last term. Question 2. If a sequence is defined by aₙ = 2 ⁿ − 1, what is a₅? A) 30 B) 31 C) 32 D) 33 Answer: B Explanation: a₅ = 2⁵ − 1 = 32 − 1 = 31. Question 3. The recursive formula aₙ₊₁ = 3aₙ + 2 with a₁ = 1 generates the sequence. What is a₃? A) 11 B) 13 C) 17 D) 20 Answer: A Explanation: a₂ = 3·1 + 2 = 5; a₃ = 3·5 + 2 = 17? Wait compute: 3·5 + 2 = 17, but answer options. Actually a₃ = 17, which is option C. Correction: Answer C.
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Explanation corrected: Using the recursion, a₂ = 5 and a₃ = 3·5 + 2 = 17, so the correct choice is C. Question 4. The factorial 5! equals: A) 20 B) 60 C) 120 D) 720 Answer: C Explanation: 5! = 5·4·3·2·1 = 120. Question 5. In an arithmetic sequence the common difference d is 4 and a₁ = 3. What is a₁₀? A) 39 B) 40 C) 41 D) 43 Answer: A Explanation: aₙ = a₁ + (n‑1)d → a₁₀ = 3 + 9 · 4 = 3 + 36 = 39. Question 6. The sum of the first n terms of an arithmetic series is given by Sₙ = n/2 (2a₁ + (n‑1)d). If a₁ = 2, d = 3, and n = 5, what is S₅? A) 35 B) 40 C) 45 D) 50
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B) |r| < 1 C) |r| > 1 D) r = − Answer: B Explanation: An infinite geometric series converges only when the absolute value of the ratio is less than 1. Question 10. Write the series 1 + 2 + 3 + … + n in sigma notation. A) ∑{k=1}^{n} k² B) ∑{k=0}^{n} k C) ∑{k=1}^{n} k D) ∑{k=1}^{n} 2k Answer: C Explanation: The sum of the first n positive integers is expressed as ∑_{k=1}^{n} k. Question 11. Which property of summation allows you to split ∑(a_k + b_k) into ∑a_k + ∑b_k? A) Distributive property B) Associative property C) Commutative property D) Linear property Answer: D Explanation: The linearity of sums states that the sum of a sum equals the sum of each part. Question 12. The principle of mathematical induction requires which two steps?
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A) Base case and contradiction B) Base case and inductive step C) Hypothesis and conclusion D) Assumption and deduction Answer: B Explanation: Induction consists of verifying the statement for the initial value (base case) and proving that truth for k implies truth for k + 1 (inductive step). Question 13. Using induction, which of the following statements is true for all n ≥ 1? A) n² + n is odd. B) n³ − n is divisible by 6. C) 2ⁿ > n². D) n! < 2ⁿ. Answer: B Explanation: n³ − n = n(n − 1)(n + 1) is the product of three consecutive integers, always divisible by 6. Question 14. What is the sum ∑_{k=1}^{4} (2k − 1)? A) 6 B) 8 C) 10 D) 12 Answer: D Explanation: Compute terms: k=1→1, k=2→3, k=3→5, k=4→7; sum = 1+3+5+7 = 16? Wait sum is 16, not among options. Let's recalc: 2k-1 gives odd numbers 1,3,5,7 sum =16. None matches. So
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D) n³ + 2n is odd. Answer: C Explanation: For n ≥ 1, factorial grows faster than 2^{n‑1}, and the inequality holds (can be shown by induction). Question 18. In a probability experiment, the sample space Ω contains 8 equally likely outcomes. What is the probability of an event containing 3 outcomes? A) 3/ B) 1/ C) 5/ D) 3/ Answer: A Explanation: Probability = number of favorable outcomes / total outcomes = 3/8. Question 19. Two events A and B are independent. Which equation must hold? A) P(A ∩ B) = P(A) + P(B) B) P(A ∩ B) = P(A) · P(B) C) P(A | B) = P(A) + P(B) D) P(A | B) = P(A) − P(B) Answer: B Explanation: Independence is defined by the product rule for joint probability. Question 20. If P(A) = 0.4, P(B) = 0.5, and A and B are mutually exclusive, what is P(A ∪ B)? A) 0. B) 0.
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C) 0.
D) 0.
Answer: C Explanation: For mutually exclusive events, P(A ∪ B) = P(A) + P(B) = 0.9? Wait 0.4+0.5=0.9, not 0.7. So answer D. Explanation corrected: Since they cannot occur together, the union probability is 0.4 + 0.5 = 0.9, thus answer D. Question 21. The conditional probability P(A | B) is defined as: A) P(A) · P(B) B) P(A ∩ B) / P(B) C) P(A ∩ B) / P(A) D) P(A) + P(B) – P(A ∩ B) Answer: B Explanation: By definition, P(A | B) = P(A ∩ B)/P(B) provided P(B) > 0. Question 22. In a binomial experiment with n = 5, p = 0.6, what is the probability of exactly k = 3 successes? A) 0. B) 0. C) 0. D) 0. Answer: B Explanation: Use C(5,3)·0.6³·0.4² = 10·0.216·0.16 = 0.3456? Actually 100.2160.16 = 0.3456. So answer A. Explanation corrected: The calculation yields 0.3456, thus answer A.
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Explanation: P(k)=e^{−λ}λ^{k}/k! → e^{−3}·3²/2! ≈ 0.2240. Question 26. Which of the following sequences is convergent? A) aₙ = (−1)ⁿ B) aₙ = n/(n+1) C) aₙ = n² D) aₙ = √n Answer: B Explanation: As n → ∞, n/(n+1) → 1, so the sequence converges to 1. Question 27. The limit lim_{n→∞} (1 + 1/n)ⁿ equals: A) e ≈ 2. B) 1 C) 0 D) ∞ Answer: A Explanation: This is the classic definition of the number e. Question 28. Which of the following statements about the harmonic series ∑_{k=1}^{∞} 1/k is true? A) It converges to 2. B) It diverges. C) It converges conditionally. D) It converges absolutely. Answer: B
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Explanation: The harmonic series diverges despite its terms tending to zero. Question 29. Using strong induction, which of the following can be proven? A) Every integer n ≥ 2 has a prime divisor. B) The sum of two even numbers is odd. C) All real numbers are rational. D) The square root of 2 is rational. Answer: A Explanation: Strong induction is suitable for statements about divisibility and prime factorization. Question 30. The principle of well‑ordering states that: A) Every non‑empty set of integers has a greatest element. B) Every non‑empty set of positive integers has a least element. C) Every set of real numbers is bounded. D) Every sequence has a monotonic subsequence. Answer: B Explanation: Well‑ordering asserts the existence of a minimal element in any non‑empty subset of ℕ. Question 31. Which of the following is the correct sigma notation for the sum of squares of the first n positive integers? A) ∑{k=1}^{n} k B) ∑{k=1}^{n} k² C) ∑{k=0}^{n} k³ D) ∑{k=1}^{n} 2k
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Answer: C Explanation: 1! + 2! + 3! = 1 + 2 + 6 = 9? Wait compute: 1+2+6=9, not 10. So answer B. Explanation corrected: The sum equals 9, thus answer B. Question 35. Which of the following statements about the binomial coefficient C(n,k) is false? A) C(n,k)=C(n,n‑k) B) C(n,0)= C) C(n,1)=n D) C(n,k)=n/k·C(n‑1,k‑1) Answer: D Explanation: The correct recurrence is C(n,k)= (n/k)·C(n‑1,k‑1) is actually valid; however the false statement is often the one that omits integer division. All given are true; need a false option. Let's replace D with C(n,k)=C(n‑1,k) + C(n‑2,k‑1). That is false. So answer D. Explanation: The correct Pascal identity is C(n,k)=C(n‑1,k)+C(n‑1,k‑1); the given D is incorrect. Question 36. In a deck of 52 cards, what is the probability of drawing an ace or a king on a single draw? A) 1/ B) 2/ C) 3/ D) 4/ Answer: B Explanation: There are 4 aces and 4 kings, total 8 favorable cards; probability = 8/52 = 2/13. Question 37. The complement rule states that P(Aᶜ) = 1 − P(A). If P(A) = 0.75, what is P(Aᶜ)?
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A) 0.
B) 0.
C) 0.
D) 0.
Answer: A Explanation: Complement probability is 1 − 0.75 = 0.25. Question 38. Which of the following is the correct expression for the variance of a Bernoulli random variable with parameter p? A) p(1 − p) B) p² C) (1 − p)² D) p + (1 − p) Answer: A Explanation: For a Bernoulli(p), Var = p(1‑p). Question 39. Using induction, prove that 2ⁿ > n for all n ≥ 1. Which step is essential? A) Show the statement for n = 0. B) Assume it holds for k and prove for k + 2. C) Assume it holds for k and prove for k + 1. D) Use contradiction. Answer: C Explanation: Standard induction requires assuming the statement for k and proving it for k + 1. Question 40. The sum of the first n odd numbers equals:
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Question 43. The expected number of heads in 10 fair‑coin tosses is: A) 4 B) 5 C) 6 D) 7 Answer: B Explanation: Expectation = n·p = 10·0.5 = 5. Question 44. Which inequality is a direct consequence of the AM‑GM inequality for two positive numbers x and y? A) (x + y)/2 ≥ √(xy) B) (x + y)/2 ≤ √(xy) C) x + y ≥ xy D) x + y ≤ xy Answer: A Explanation: AM‑GM states the arithmetic mean is at least the geometric mean. Question 45. In a Markov chain with transition matrix [ P=\begin{pmatrix}0.7&0.3\0.4&0.6\end{pmatrix}, ] what is the probability of moving from state 1 to state 2 in two steps? A) 0. B) 0. C) 0.
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D) 0.
Answer: C Explanation: Compute (P²)₁₂ = 0.7·0.3 + 0.3·0.6 = 0.21 + 0.18 = 0.39? Wait correct multiplication: Row1·Column2 = 0.70.3 + 0.30.6 = 0.21 + 0.18 = 0.39. Not among options. Replace options: A) 0.39 B) 0.45 C) 0.51 D) 0.57. Answer A. Explanation: Two‑step probability is entry (1,2) of P², which equals 0.39. Question 46. Which of the following statements about the Central Limit Theorem (CLT) is true? A) The sum of any finite number of random variables is normal. B) The sample mean of a large i.i.d. sample approximates a normal distribution regardless of the original distribution. C) CLT only applies to Bernoulli trials. D) CLT requires the population variance to be zero. Answer: B Explanation: CLT states that the distribution of the sample mean approaches normality as sample size grows. Question 47. If X ~ Uniform(0,1), what is P(0.2 < X < 0.8)? A) 0. B) 0. C) 0. D) 0. Answer: C Explanation: Length of interval = 0.8 − 0.2 = 0.6.
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Question 51. Which of the following series is conditionally convergent? A) ∑{n=1}^{∞} 1/n² B) ∑{n=1}^{∞} (−1)^{n+1}/n C) ∑{n=1}^{∞} 1/n D) ∑{n=1}^{∞} (−1)^{n}/n² Answer: B Explanation: The alternating harmonic series converges, but the absolute series diverges, making it conditionally convergent. Question 52. If aₙ = (−1)^{n}/n, what is lim_{n→∞} aₙ? A) 1 B) − C) 0 D) Does not exist Answer: C Explanation: The magnitude 1/n → 0, and the sign alternates, so limit is 0. Question 53. In a geometric distribution with success probability p = 0.2, what is the probability that the first success occurs on the 4th trial? A) 0. B) 0. C) 0. D) 0. Answer: B
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Explanation: P(X=4) = (1‑p)^{3}·p = 0.8³·0.2 = 0.512·0.2 = 0.1024? Actually 0.512*0.2 = 0.1024, which is option A. So answer A. Explanation corrected: The correct probability is 0.1024. Question 54. Which of the following is the correct expression for the probability mass function of a hypergeometric distribution? A) C(K, k)·C(N‑K, n‑k) / C(N, n) B) C(N, k)·p^{k}·(1‑p)^{N‑k} C) λ^{k}e^{−λ}/k! D) 1/(n+1) Answer: A Explanation: Hypergeometric PMF counts ways to choose k successes from K and n‑k failures from N‑K. Question 55. The sum of the interior angles of a convex n‑gon is: A) (n‑2)· 180 ° B) n·180° C) (n‑1)· 180 ° D) (n‑2)· 90 ° Answer: A Explanation: Formula for interior angle sum of a polygon with n sides. Question 56. Which of the following statements about the law of total probability is correct? A) It requires events to be mutually exclusive and exhaustive. B) It only applies to independent events.
