AMC 12 AMC 12 Sequences, Series & Limits 2 — Questions and Answers
Question 1: What is lim(x→∞) of (3x² + 1)/(x² − 2)?
- 0
- 1
- 3 (Correct answer)
- ∞
Correct answer: 3
Dividing numerator and denominator by x² yields (3 + 1/x²)/(1 − 2/x²) → 3 as x → ∞.
Question 2: What is the sum 1/(1·2) + 1/(2·3) + 1/(3·4) + … + 1/(n(n+1))?
- 1
- n/(n+1) (Correct answer)
- 1/(n+1)
- n(n+1)/2
Correct answer: n/(n+1)
Partial fractions give 1/(k(k+1)) = 1/k − 1/(k+1), and the series telescopes to 1 − 1/(n+1) = n/(n+1).
Question 3: The Fibonacci sequence is defined by F₁ = 1, F₂ = 1, Fₙ = Fₙ₋₁ + Fₙ₋₂. What is F₇?
- 8
- 11
- 13 (Correct answer)
- 21
Correct answer: 13
Computing in order: F₃=2, F₄=3, F₅=5, F₆=8, F₇=13.
Question 4: What is lim(x→0) of sin(x)/x?
- 0
- ∞
- 1 (Correct answer)
- undefined
Correct answer: 1
This is a standard fundamental limit: lim(x→0) sin(x)/x = 1.
Question 5: An arithmetic series has 10 terms with first term 1 and total sum 100. What is the common difference?
- 1
- 2 (Correct answer)
- 3
- 4
Correct answer: 2
S = n/2·(2a₁ + (n−1)d) gives 100 = 5(2 + 9d), so 9d = 18 and d = 2.
Question 6: Which formula gives the sum of the first n perfect squares?
- n(n+1)/2
- n(n+1)(2n+1)/6 (Correct answer)
- n²(n+1)²/4
- n(2n+1)(2n−1)/3
Correct answer: n(n+1)(2n+1)/6
The sum 1² + 2² + … + n² = n(n+1)(2n+1)/6.
AMC 12 AMC 12 Sequences, Series & Limits 2
What is lim(x→∞) of (3x² + 1)/(x² − 2)?