TADM2E 1.15

<b>Step 1:</b> Show that the statement holds for the basis case <math>n = 1</math><br>

<math>\frac {1}{i(i+1)} = \frac {n}{n+1}</math><br><br>

<math>\frac {1}{1(1+1)} = \frac {1}{1+1}</math><br><br>

<math>\frac {1}{2} = \frac {1}{2}</math><br><br>

Since <math>1/2 = 1/2</math>, the basis case is true.<br><br>

<b>Step 2:</b> Assume that that summation is true up to n.<br><br> <b>Step 3:</b> Show that on the assumption that the summation is true for n, it follows that it is true for n + 1.

<math>\sum_{i = 1}^{n+1} = \frac{n+1}{n+1+1} = \frac{n}{n+1} + \frac{1}{(n+1)(n+1+1)}</math><br> <math>\frac{n+1}{n+2} = \frac{n(n+2)}{(n+1)(n+2)} + \frac{1}{(n+1)(n+2)}</math><br> <math>\frac{n+1}{n+2} = \frac{n(n+2)+1}{(n+1)(n+2)}</math><br> <math>\frac{n+1}{n+2} = \frac{n^2+2n+1}{(n+1)(n+2)}</math><br> <math>\frac{n+1}{n+2} = \frac{(n+1)(n+1)}{(n+1)(n+2)}</math><br> <math>\frac{n+1}{n+2} = \frac{(n+1)}{(n+2)}</math><br>

QED