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Goldman Sachs·Software Engineer·Technical Phone Screen·Intermediate

Intermediate
Jun 2026

Summary

Interviewed for a data science role at Goldman Sachs and got a math question I did not expect to see outside of a pure math class.

Questions Asked (1)

Q1

As the Fibonacci sequence progresses, what value does the ratio of a term to its preceding term converge to?

Algorithms & Data Structures
Author's notes

Blanked for a second because I was expecting something about ML or statistics.

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AI HintsAI Generated

Suggested Approach

State the answer directly: the ratio converges to the golden ratio, approximately 1.618. Then briefly explain why by deriving the limit from the recurrence relation and connecting it to the characteristic equation x^2 = x + 1.

Pro tip: Mention that the convergence is oscillatory and that the ratio alternates above and below the golden ratio, which shows deeper understanding and can impress interviewers at quantitative firms like Goldman Sachs.

1. State the answer

Clearly say that the ratio converges to the golden ratio, φ ≈ 1.6180339887.

2. Set up the limit

Assume the ratio F(n+1)/F(n) converges to L. Then as n→∞, F(n+2)/F(n+1) also converges to L.

3. Derive the equation

Use the Fibonacci recurrence F(n+2) = F(n+1) + F(n) to write L = 1 + 1/L, leading to L^2 - L - 1 = 0.

4. Solve and interpret

Solve the quadratic to get L = (1 ± √5)/2. Since the ratio is positive, L = (1 + √5)/2 ≈ 1.618.

5. Discuss convergence

Note that the convergence is oscillatory and that the ratio alternates around φ, but the magnitude of the difference decreases geometrically.

Key Points to Mention

  • The golden ratio φ = (1 + √5)/2 ≈ 1.618
  • Derivation from the recurrence relation F(n+2) = F(n+1) + F(n)
  • The characteristic equation x^2 = x + 1
  • The ratio alternates above and below φ as it converges
  • Connection to Binet's formula and the closed-form expression for Fibonacci numbers
  • Practical relevance: appears in algorithm analysis (e.g., Fibonacci heap) and nature

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.