Computation Workshop Solution Checker

Two Parallel Recurrences

Two sequences \(a(0), a(1), a(2), \ldots \) and \(b(0), b(1), b(2), \ldots \) are defined by:

  • \(a(0) = b(0) = 0\),
  • \(a(n+1) = a(n) + b(n) + 1\), and
  • \(b(n+1) = a(n) + 2b(n) + 3\).

Part A What is \(a(10)\)?
Part B What is the remainder when \(a(2021)\) is divided by \(10^9+7\)?
Part C What is the remainder when \(a(1234567890)\) is divided by \(10^9+7\)?

Inverse Harmonic Function

The Harmonic Series \(H(n)\) is defined to be the sum of the reciprocals of the first \(n\) positive integers. \( H(n) = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots + \frac{1}{n} \). The Inverse Harmonic Function \(I(x)\) is defined to be the smallest positive integer \(n\) such that \(H(n) \geq x\). So for example \(I(2.0) = 4\) because \(H(4) > 2.0\) but \(H(3) < 2.0\).

Part A What is \(I(4)\)?
Part B What is \(I(18)\)?