Resolved: How can I efficiently calculate the mathematical limit for average number of rounds in a game?

// LIMIT could be any number, the larger it is, the more accurate the result.<br />
const LIMIT = 100;<br />
// r is the number of rounds left if at least 1 of the sides hit<br />
// x is the chance that both sides miss and the round count gets increased,<br />
// but the battle state stays the same.<br />
function approximateLimitForNumberOfRounds(r: number, x: number) {<br />
  let approx = r / (1 - x);<br />
  // n -> infinity<br />
  for (let n = 1; n < LIMIT; n++) {
    approx += x ** n;
  }
  return approx;
}
[/code]

</p>



How can I modify this function to exactly calculate the number of rounds left, instead of approximating it? (noting that since <code>x</code> is a chance, it is contained in <code>(0, 1)</code> or <code>0 < x < 1</code>).<br> 



<h2 class="wp-block-heading">Answer:</h2>



We can note that <code>approx</code> takes on the following values:<br> 


<p>[code]r / (1 - x) # I refer to this as 'a' below<br />
a + x<br />
a + x + x^2<br />
a + x + x^2 + x^3<br />
a + x + x^2 + ... + x^n

a + (the sum of x^k from k = 1 to k = n)

x + x^2 + x^3 + ... x^inf = x/(1-x)

approx = r / (1 - x) + x / (1 - x)
approx = (r + x) / (1 - x)