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GRE Sequences, Probability and Counting Theory — practice questions

Explore 22 free Sequences, Probability and Counting Theory questions in GRE Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Sequences, Probability and Counting Theory questions and exercises with answers

Showing questions 1–22 of 22.

A sequence is defined by the explicit formula a_n = -3n + 8. Which term of the sequence equals -22?Read the answer and worked solution →A sequence is defined recursively by a_1 = 3 and a_n = 2a_(n-1) - 1 for n >= 2. What is the fifth term?Read the answer and worked solution →A sequence is defined by a_n = (n + 1)!. What is its fourth term?Read the answer and worked solution →In the Fibonacci sequence each term is the sum of the two preceding terms. The eighth term is 21 and the ninth term is 34. What is the eleventh term?Read the answer and worked solution →How many terms are in the finite arithmetic sequence 5, -2, -9, ..., -65?Read the answer and worked solution →A stadium has 10 seats in the first row, 13 in the second, 16 in the third, and the pattern continues for all 56 rows. What is the total seating capacity?Read the answer and worked solution →A new analyst earns $30,000 a year with a promised 3% raise at the end of every year. What is the annual salary after two raises?Read the answer and worked solution →A geometric sequence has first term 2 and fourth term 54. What is its third term?Read the answer and worked solution →Evaluate the geometric series given by the sum of 3(2^k) for k = 1 to 5.Read the answer and worked solution →An infinite geometric series has first term 9 and common ratio 4/3. What can be said about its sum?Read the answer and worked solution →The repeating decimal 0.777... can be written as the geometric series 0.7 + 0.07 + 0.007 + ... . Which fraction equals this decimal?Read the answer and worked solution →Twelve sprinters enter a race. In how many different ways can the gold, silver, and bronze medals be awarded?Read the answer and worked solution →A sundae bar offers 6 toppings, and any number of them may be chosen, including none. How many different sundaes are possible?Read the answer and worked solution →How many distinct rearrangements can be made from the letters of the word CARRIER?Read the answer and worked solution →Two six-sided number cubes are rolled and the faces are summed. What is the probability that the sum is less than 10?Read the answer and worked solution →A six-sided number cube is rolled once. What is the probability of rolling a number greater than 2?Read the answer and worked solution →Match each sequence idea to its ruleRead the answer and worked solution →Sort each sequence by its patternRead the answer and worked solution →Count the waysRead the answer and worked solution →Complete the formulasRead the answer and worked solution →Judge the probability claimsRead the answer and worked solution →Pick the bigger countRead the answer and worked solution →

Worked practice questions

A sequence is defined by the explicit formula a_n = -3n + 8. Which term of the sequence equals -22?

  1. The 9th term
  2. The 10th term
  3. The 8th term
  4. The 11th term
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Correct answer: B. The 10th term

1. An explicit formula gives any term directly from its position n, so set the formula equal to -22. 2. Solve -3n + 8 = -22: subtract 8 from both sides to get -3n = -30. 3. Divide by -3 to get n = 10, so the 10th term equals -22. 4. Checking n = 9 gives -19 and n = 11 gives -25, so neither neighboring position works. _Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 13 "Sequences, Probability and Counting Theory", section 13.1 SEQUENCES AND THEIR NOTATIONS_

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A sequence is defined recursively by a_1 = 3 and a_n = 2a_(n-1) - 1 for n >= 2. What is the fifth term?

  1. 31
  2. 17
  3. 33
  4. 35
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Correct answer: C. 33

1. A recursive rule builds each term from the one before it, starting from a_1 = 3. 2. a_2 = 2(3) - 1 = 5 and a_3 = 2(5) - 1 = 9. 3. a_4 = 2(9) - 1 = 17. 4. a_5 = 2(17) - 1 = 33. 5. Option B, 17, is the fourth term, the classic off-by-one trap when iterating a recursion. _Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 13 "Sequences, Probability and Counting Theory", section 13.1 SEQUENCES AND THEIR NOTATIONS_

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A sequence is defined by a_n = (n + 1)!. What is its fourth term?

  1. 120
  2. 24
  3. 720
  4. 60
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Correct answer: A. 120

1. Substitute n = 4 into the formula: a_4 = (4 + 1)! = 5!. 2. Expand the factorial: 5! = 5 x 4 x 3 x 2 x 1 = 120. 3. Option B, 24, is 4!, which comes from forgetting to add 1 to n before taking the factorial. _Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 13 "Sequences, Probability and Counting Theory", section 13.1 SEQUENCES AND THEIR NOTATIONS_

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