Mathematical Induction

When n = 1 ,
1 / (1x3x5) = 1 / [(2-1)(2+1)(2+3)] is true.Assume that when n = k the statement is true , i.e.1/1X3X5+1/3X5X7+1/5X7X9+...1/(2k-1)(2k+1)(2k+3) = k(k+2) / 3(2k+1)(2k+3)When n = k+1 :1/1X3X5+1/3X5X7+1/5X7X9+...+1/(2k-1)(2k+1)(2k+3) +
1 / {[(2(k+1)-1] [2(k+1)+1] [2(k+1)+3]}= k(k+2) / [3(2k+1)(2k+3)] + 1 / {[(2(k+1)-1] [2(k+1)+1] [2(k+1)+3]}= k(k+2) / [3(2k+1)(2k+3)] + 1 / [(2k+1)(2k+3)(2k+5)]= [k(k+2)(2k+5) + 3] / [3(2k+1)(2k+3)(2k+5)] = [k(k+2)(2k+1 + 4) + 3] / [3(2k+1)(2k+3)(2k+5)] = [(2k + 1)(k^2 + 2k) + 4k^2 + 8k + 3] / [3(2k+1)(2k+3)(2k+5)] = [(2k + 1)(k^2 + 2k) + (2k + 1)(2k + 3)] / [3(2k+1)(2k+3)(2k+5)] = (k^2 + 2k + 2k + 3) / [3(2k+3)(2k+5)] = (k+1)(k+3) / [3(2k+3)(2k+5)] = (k+1)(k+1 + 2) / { [3(2(k+1)+1] [2(k+1)+3] } When n = k is true , n = k+1 is also true.By Mathematical Induction it is true for all positive integers n.