Can someone please show me how to work these two questions? I am having major problems with even getting started. 1) Use mathematical induction to prove the truth of the following: 1 / 1*2 + 1 / 2*3 + 1 / 3*4+...+1/n(n + 1) = n/ n + 1 2) Use mathematical induction to prove the truth of the following: 12 + 22 + 32 + ... + n 2 = n(n + 1)(2n + 1) / 6
show that it works when n = 1 1/1*2 = 1/2 then show that if in works for n then it will also work for n+1 1 / 1*2 + 1 / 2*3 + 1 / 3*4+...+1/n(n + 1) = n/ n + 1 1 / 1*2 + 1 / 2*3 + 1 / 3*4+...+1/n(n + 1) + 1/(n+1)(n+2) = n/ (n + 1) + 1/(n+1)(n+2) = (n(n+2) + 1)/(n+1)(n+2) = (n^2+2n + 1)/(n+1)(n+2) = (n+ 1)^2/(n+1)(n+2) = (n+ 1)/(n+2) 2) 12+22+32? 1^2 + 2^2 + 3^2...n^2 = n(n + 1)(2n + 1) / 6 if n = 1 1^2 = 1(1+1)(2+1)/6 = 1 1^2 + 2^2 + 3^2...n^2 = n(n + 1)(2n + 1) / 6 1^2 + 2^2 + 3^2...n^2 + (n+1)^2 = n(n + 1)(2n + 1) / 6 + (n+1)^2 (2n^3 + 3n^2 + n)/6 + n^2 + 2n + 1 (2n^3 + 9n^2 + 13n + 6)/6 now you scratch your head.. but you know where you are trying to get to... so...factor out n+1 (n+1)(2n^2 + 7n + 6)/6 (n+1)(n+2)(2n+3)/6 (n+1)((n+1)+1)(2(n+1)+1)/6