NBT NBT Mathematics: Sequences, Series and Patterns 3 — Questions and Answers
Question 1: The 7th term of an arithmetic sequence is 43 and the common difference is 6. What is the first term?
- 5
- 7 (Correct answer)
- 9
- 11
Correct answer: 7
T₇ = a + 6d = a + 36 = 43 → a = 7.
Question 2: For which value(s) of n does the series ∑ₖ₌₁ⁿ k = 55?
- n = 9
- n = 10 (Correct answer)
- n = 11
- n = 12
Correct answer: n = 10
Sₙ = n(n+1)/2 = 55 → n(n+1) = 110 → n = 10.
Question 3: A pattern of dots: Row 1 has 1 dot, Row 2 has 3, Row 3 has 5. How many dots are in Row n?
- 2n
- 2n − 1 (Correct answer)
- n² − 1
- n + 2
Correct answer: 2n − 1
Each row adds 2 more dots; the nth row has 2n − 1 dots.
Question 4: A geometric series has S∞ = 12 and r = 1/3. What is the first term?
- 6
- 8 (Correct answer)
- 9
- 10
Correct answer: 8
S∞ = a/(1−r) = a/(2/3) = 3a/2 = 12 → a = 8.
Question 5: The second differences of a sequence are all equal to 6. If T₁ = 2 and T₂ = 5, what is T₃?
- 10
- 12
- 14 (Correct answer)
- 16
Correct answer: 14
First difference Δ₁ = T₂ − T₁ = 3; next first difference = 3 + 6 = 9; T₃ = 5 + 9 = 14.
Question 6: Which sequence is neither arithmetic nor geometric?
- 1, 4, 7, 10, 13
- 2, 4, 8, 16, 32
- 1, 2, 4, 7, 11 (Correct answer)
- 5, 10, 15, 20, 25
Correct answer: 1, 2, 4, 7, 11
1, 2, 4, 7, 11 has increasing first differences (1, 2, 3, 4) so it is quadratic, not arithmetic or geometric.
Question 7: Simplify: ∑ₖ₌₁⁶ (3k + 2)
- 74
- 75 (Correct answer)
- 77
- 80
Correct answer: 75
Sum = 3∑k + 2(6) = 3(21) + 12 = 63 + 12 = 75. Wait: 5+8+11+14+17+20 = 75.
NBT NBT Mathematics: Sequences, Series and Patterns 3
The 7th term of an arithmetic sequence is 43 and the common difference is 6.
What is the first term?