PRE CALCULUS Pre-Calculus Sequences and Series 3 — Questions and Answers
Question 1: In a geometric sequence, a₃ = 12 and a₆ = 96. What is the common ratio?
- 1.5
- 2 (Correct answer)
- 3
- 4
Correct answer: 2
a₆/a₃ = r³ = 96/12 = 8, so r = 2.
Question 2: What is the sum of the first 100 positive integers?
- 4950
- 5000
- 5050 (Correct answer)
- 5100
Correct answer: 5050
Using Gauss's formula S = n(n+1)/2 = 100(101)/2 = 5050.
Question 3: A sequence is defined by aₙ = n² − n. Which term equals 20?
- 4th
- 5th (Correct answer)
- 6th
- 7th
Correct answer: 5th
n² − n = 20 → n(n−1) = 20 → n = 5, so the 5th term equals 20.
Question 4: The sum of a finite arithmetic series with 20 terms is 500. If the first term is 5, what is the last term?
- 40
- 45 (Correct answer)
- 50
- 55
Correct answer: 45
S = n(a₁ + aₙ)/2 → 500 = 20(5 + aₙ)/2 → aₙ = 45.
Question 5: Which sequence is neither arithmetic nor geometric?
- 2, 4, 6, 8, …
- 3, 9, 27, 81, …
- 1, 4, 9, 16, … (Correct answer)
- 5, 10, 15, 20, …
Correct answer: 1, 4, 9, 16, …
1, 4, 9, 16, … are perfect squares; differences grow, so it is neither arithmetic nor geometric.
Question 6: If the 5th partial sum of a series is 62 and the 4th partial sum is 50, what is the 5th term?
- 10
- 12 (Correct answer)
- 14
- 16
Correct answer: 12
The 5th term = S₅ − S₄ = 62 − 50 = 12.
Question 7: For the series Σ (from n=1 to ∞) of (1/3)ⁿ, what is the sum?
- 1/4
- 1/3
- 1/2 (Correct answer)
- 2/3
Correct answer: 1/2
S = (1/3)/(1 − 1/3) = (1/3)/(2/3) = 1/2.
PRE CALCULUS Pre-Calculus Sequences and Series 3
In a geometric sequence, a₃ = 12 and a₆ = 96.
What is the common ratio?