NBT NBT Mathematics: Sequences, Series and Patterns 4 — Questions and Answers
Question 1: The sum of the first n terms of a series is given by Sₙ = 3n² − n. What is T₄?
- 20 (Correct answer)
- 21
- 22
- 24
Correct answer: 20
T₄ = S₄ − S₃ = (3×16−4) − (3×9−3) = 44 − 24 = 20.
Question 2: A geometric sequence has T₂ = 6 and T₅ = 162. What is the first term?
- 2 (Correct answer)
- 3
- 4
- 6
Correct answer: 2
T₅/T₂ = r³ = 162/6 = 27 → r = 3; T₁ = T₂/r = 6/3 = 2.
Question 3: How many terms of the series 8 + 11 + 14 + … are needed for the sum to first exceed 100?
- 6
- 7 (Correct answer)
- 8
- 9
Correct answer: 7
Sₙ = n/2(16 + 3(n−1)) > 100. For n=7: S₇ = 7/2(37) = 129.5 > 100; S₆ = 3(32) = 96 < 100.
Question 4: Express 0.\overline{36} as a fraction in lowest terms.
- 4/11 (Correct answer)
- 9/25
- 36/99
- 4/11
Correct answer: 4/11
0.363636… = 36/99 = 4/11.
Question 5: The nth term of a sequence is Tₙ = n(n + 2). Which of the following is NOT a term of this sequence?
- 10 (Correct answer)
- 8
- 15
- 24
Correct answer: 10
Tₙ = n(n+2) gives 3, 8, 15, 24 for n=1,2,3,4. The value 10 is between T₂=8 and T₃=15 and cannot be produced.
Question 6: If the first term of a geometric sequence is 81 and the common ratio is 1/3, what is S∞?
- 108.5
- 120
- 121.5 (Correct answer)
- 135
Correct answer: 121.5
S∞ = 81/(1 − 1/3) = 81/(2/3) = 81 × 3/2 = 121.5.
Question 7: Determine the value of k if 2k, k + 3, and k − 1 are three consecutive terms of an arithmetic sequence.
- k = −1
- k = 1 (Correct answer)
- k = 4
- k = 7
Correct answer: k = 1
(k+2)² = k(k+8) → k²+4k+4 = k²+8k → 4k = 4 → k = 1. Verify: 1, 3, 9 has r=3. ✓
NBT NBT Mathematics: Sequences, Series and Patterns 4
The sum of the first n terms of a series is given by Sₙ = 3n² − n.
What is T₄?