AMC 12 AMC 12 Sequences, Series & Limits 2

Question 1

What is lim(x→∞) of (3x² + 1)/(x² − 2)?

Accepted Answer

3

Answer

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))?

Accepted Answer

n/(n+1)

Answer

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?

Accepted Answer

13

Answer

Computing in order: F₃=2, F₄=3, F₅=5, F₆=8, F₇=13.

Question 4

What is lim(x→0) of sin(x)/x?

Accepted Answer

1

Answer

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?

Accepted Answer

2

Answer

S = n/2·(2a₁ + (n−1)d) gives 100 = 5(2 + 9d), so 9d = 18 and d = 2.

AMC 12 - - American Mathematics Contest 12 Practice Test