Binet's formula states F_n = (φⁿ − ψⁿ)/√5 where φ = (1+√5)/2 and ψ = (1−√5)/2 are both irrational. A student claims this formula must be wrong because it produces irrational outputs for integer inputs. What is the correct explanation?
AThe formula only works for even values of n, where the irrationals happen to cancel
Bφ and ψ are actually rational numbers that merely resemble surds in written form
CThe irrational parts always cancel exactly when φⁿ and ψⁿ are subtracted and divided by √5, producing an integer
DThe formula is an approximation that rounds to the nearest integer rather than giving an exact result
Binet's formula is exact, not approximate. The key is that φ and ψ are conjugate surds: when raised to the same power and subtracted, the irrational √5 terms cancel in just the right way to leave an integer. Option D conflates Binet's formula with the useful approximation that F_n ≈ φⁿ/√5 (which works because |ψⁿ/√5| < 0.5 and can be ignored). The exact formula always gives an integer; rounding is not needed.
Question 2 Multiple Choice
According to the divisibility property gcd(F_m, F_n) = F_{gcd(m,n)}, which of the following is true?
AF_3 = 2 divides every Fibonacci number because 3 divides every index
BF_5 = 5 divides F_9 because F_5 < F_9
CF_4 = 3 divides F_12 because gcd(4, 12) = 4, so F_4 | F_12
Dgcd(F_6, F_9) = F_3 = 2, so F_6 and F_9 share no common factor greater than 2
The property states gcd(F_m, F_n) = F_{gcd(m,n)}, and a direct corollary is that F_m divides F_n if and only if m divides n. Since 4 divides 12, F_4 = 3 divides F_12 = 144. Option A is wrong because 3 does not divide every integer index — only every third Fibonacci number is divisible by F_3 = 2. Option B confuses size with divisibility. Option D correctly computes gcd(F_6, F_9) = F_{gcd(6,9)} = F_3 = 2, but then misstates the conclusion — it means they share a common factor of exactly 2.
Question 3 True / False
Since |ψ| = |(1−√5)/2| < 1, the term ψⁿ/√5 approaches zero as n grows, meaning every Fibonacci number equals the nearest integer to φⁿ/√5.
TTrue
FFalse
Answer: True
Because |ψ| ≈ 0.618 < 1, ψⁿ shrinks exponentially. For all n ≥ 0, |ψⁿ/√5| < 0.5, so F_n = (φⁿ − ψⁿ)/√5 must be the nearest integer to φⁿ/√5. This gives a practical way to compute large Fibonacci numbers: raise φ to the power n, divide by √5, and round.
Question 4 True / False
The ratio of consecutive Fibonacci numbers F_{n+1}/F_n converges to √5 as n increases.
TTrue
FFalse
Answer: False
The ratio converges to φ = (1+√5)/2 ≈ 1.618, not √5 ≈ 2.236. This follows directly from Binet's formula: F_{n+1}/F_n = (φⁿ⁺¹ − ψⁿ⁺¹)/(φⁿ − ψⁿ) → φ as the ψ terms vanish. The ratio alternates above and below φ, converging from both sides.
Question 5 Short Answer
Explain the tiling interpretation of Fibonacci numbers and how it can be used to prove the identity F_1 + F_2 + ··· + F_n = F_{n+2} − 1.
Think about your answer, then reveal below.
Model answer: A 1×n board can be tiled using 1×1 tiles and 1×2 dominoes. For the first square: either a 1×1 tile covers it (leaving n−1 squares, F_n tilings) or a 1×2 domino covers the first two (leaving n−2 squares, F_{n−1} tilings). This gives the Fibonacci recurrence, so the count is F_{n+1} tilings for a board of length n. To prove the sum identity, count tilings of a 1×(n+2) board by where the last tile or domino ends: if it ends at position k, the remaining board has F_k tilings. Summing over all possible endings gives F_1 + F_2 + ··· + F_{n+1} = F_{n+3} − 1, which after reindexing yields the stated identity. The tiling interpretation converts algebraic identities into combinatorial counting arguments.
The key power of the tiling interpretation is that it makes algebraic Fibonacci identities visually obvious by counting the same configurations two different ways — a standard combinatorial proof technique called double counting.