NBT NBT Mathematics: Sequences, Series and Patterns 5 — Questions and Answers
Question 1: What is the sum of all integers from 1 to 100?
- 4950
- 5000
- 5050 (Correct answer)
- 5100
Correct answer: 5050
Sₙ = n(n+1)/2 = 100(101)/2 = 5050.
Question 2: A sequence has Tₙ = 5 × 2ⁿ⁻¹. Which statement is true?
- It is arithmetic with d = 2
- It is geometric with r = 2 (Correct answer)
- It is geometric with r = 5
- It is arithmetic with d = 5
Correct answer: It is geometric with r = 2
The formula Tₙ = arⁿ⁻¹ with a = 5 and r = 2 defines a geometric sequence.
Question 3: If ∑ₙ₌₁⁸ aₙ = 96 and each term is equal, what is each term?
- 8
- 10
- 12 (Correct answer)
- 16
Correct answer: 12
If all 8 terms are equal, each term = 96 ÷ 8 = 12.
Question 4: The 4th term of an arithmetic sequence is 25 and the 9th term is 50. What is the common difference?
- 4
- 5 (Correct answer)
- 6
- 7
Correct answer: 5
T₉ − T₄ = 5d = 25 → d = 5.
Question 5: A pattern of squares: 1st figure has 1 square, 2nd has 5, 3rd has 13, 4th has 25. What is the general term?
- Tₙ = 4n − 3
- Tₙ = 2n² − 2n + 1 (Correct answer)
- Tₙ = n² + 2n − 2
- Tₙ = 3n² − 2n
Correct answer: Tₙ = 2n² − 2n + 1
Tₙ = 2n²−2n+1 gives T₁=1, T₂=5, T₃=13, T₄=25, matching the pattern exactly.
Question 6: Which value of r makes the infinite geometric series ∑ arⁿ diverge?
- r = 0.5
- r = −0.8
- r = 1.2 (Correct answer)
- r = −0.99
Correct answer: r = 1.2
An infinite geometric series converges only when |r| < 1; r = 1.2 has |r| > 1 so it diverges.
Question 7: The first term of a sequence is 64 and each subsequent term is half the previous. What is the sum of the first 6 terms?
- 124
- 126 (Correct answer)
- 127
- 128
Correct answer: 126
S₆ = 64(1 − (1/2)⁶)/(1 − 1/2) = 64(63/64)/(1/2) = 63 × 2 = 126.
NBT NBT Mathematics: Sequences, Series and Patterns 5
What is the sum of all integers from 1 to 100?