Question 1183960
Step 1.  For n = 1, {{{1/(1*2*3) = (1*4)/(4*2*3)}}}, and the statement is true.


Step 2. (Inductive Hypothesis)  Let statement be true for n = k for some positive integer k, that is,

{{{1/(1*2*3) + 1/(2*3*4) + 1/(3*4*5) +"..."+ 1/(k(k+1)(k+2)) = (k(k+3))/(4(k+1)(k+2))}}}


Step 3. Prove statement for n = k+1: {{{ (1/(1*2*3) + 1/(2*3*4) + 1/(3*4*5) + "..."+ 1/(k(k+1)(k+2)) ) + 1/((k+1)(k+2)(k+3)) = (k(k+3))/(4(k+1)(k+2)) +1/((k+1)(k+2)(k+3)) }}}.



= {{{(1/((k+1)(k+2)))*((k(k+3))/4+1/(k+3)) =(1/((k+1)(k+2)))*( (k(k+3)^2+4)/(4(k+3)) ) )}}}


= {{{((k+4)(k+1)^2)/(4(k+1)(k+2)(k+3)) = ((k+1)(k+4))/(4(k+2)(k+3))}}}


Hence the staement is true for n = k+1.  Therefore the staement is true for {{{n >= 1}}}.