Gusty Games

 

Computation Workshop Solution Checker


The Fibonacci Sequence

The sequence \(1, 1, 2, 3, 5, \ldots\) is called the Fibonacci Sequence. It is the sequence \(f(1)\), \(f(2)\), \(f(3)\), \(f(4)\), \(\ldots\) defined by

  • \(f(1) = 1\),
  • \(f(2) = 1\), and
  • \(f(n + 2) = f(n + 1) + f(n)\) for all \(n \ge 1\).

So for example \(f(12) = 144\). So the first two digits of \(f(12)\) are \(14\) and the last two digits of \(f(12)\) are \(44\).

Part A What are the first three digits of \(f(123)\)?
Part B What are the first six digits of \(f(123456)\)?
Part C What are the last six digits of \(f(123456)\)?
Part D What are the first nine digits of \(f(123456789)\)?
Part E What are the last nine digits of \(f(123456789)\)?

Digital Self Powers

The self-power of a number \(x\) is defined to be \(x^x\). The digital-self-power of a number is defined to be the sum of the self powers of all of its digits (in base 10). For example, the digital-self-power of \(402\) is \(4^4 + 0^0 + 2^2 = 256 + 1 + 4 = 261\).

Part A Find an integer which is one more than its digital-self-power.
Part B Find an integer (greater than 1) which equals its digital-self-power
Part C Find a positive integer which is one less than its digital-self-power
Part D Find the largest integer which is at most one away from its digital-self-power