Functions and Sequences Flashcards

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

Answer: 31

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

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

Answer: 27

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

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

Answer: 26

a₃=6, a₄=10, a₅=16, a₆=26.

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

Answer: 2/3

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

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)?

Answer: 3n − 1

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

If f(x) = x² + 1 and a sequence is defined by a₁ = 0, aₙ₊₁ = f(aₙ), what is a₃?

Answer: 2

a₂ = f(0) = 0²+1 = 1; a₃ = f(1) = 1²+1 = 2.

A sequence satisfies a₁ = 2 and aₙ₊₁ = aₙ / (1 + aₙ). What is a₄?

Answer: 2/7

a₂ = 2/3, a₃ = (2/3)/(5/3) = 2/5, a₄ = (2/5)/(7/5) = 2/7.