COMC Functions and Sequences 3

Question 1

A sequence satisfies a₁ = 1 and aₙ₊₁ = aₙ + n² for n ≥ 1. What is a₅?

Accepted Answer

31

Answer

a₂=1+1=2, a₃=2+4=6, a₄=6+9=15, a₅=15+16=31.

Question 2

If f(n) = n(n+1)/2 (the nth triangular number), what is f(10) − f(7)?

Accepted Answer

27

Answer

f(10) = 55, f(7) = 28, so f(10) − f(7) = 27.

Question 3

A sequence satisfies a₁ = 2, a₂ = 4, and aₙ = aₙ₋₁ + aₙ₋₂ for n ≥ 3. What is a₆?

Accepted Answer

26

Answer

a₃=6, a₄=10, a₅=16, a₆=26.

Question 4

An infinite geometric series with first term a has sum 24. If a = 8, what is the common ratio r?

Accepted Answer

2/3

Answer

S = a/(1−r) = 24 with a = 8 gives 1−r = 8/24 = 1/3, so r = 2/3.

Question 5

A function on positive integers satisfies f(1) = 2 and f(n) = f(n−1) + 3 for n ≥ 2. Which closed form equals f(n)?

Accepted Answer

3n − 1

Answer

The arithmetic sequence has first term 2 and common difference 3, giving f(n) = 2 + 3(n−1) = 3n − 1.

COMC - Canadian Open Mathematics Challenge Practice Test