Mathematical induction333

Prove, by mathematical induction, that for all positive integers n,
1x2 + 2x3 + 3x4 +...+ n(n+1) = (1/3) n (n+1) (n+2)
Solution :
When n=1 , 1x2 = (1/3)(1)(1+1)(1+2) , it is true for n=1 ,
Assume it is true for n=k , where k is a positive integer ,

i.e. 1x2+2x3+...+k(k+1)= (1/3)(k)(k+1)(k+2)
Consider n =k+1 ,

1x2+2x3+...+k(k+1)+(k+1)(k+2)
=(1/3)(k)(k+1)(k+2) + (k+1)(k+2)
=(k+1)(k+2)(1/3k+1)
=(1/3)(k+1)(k+2)(k+3) , it is true for n=k+1
By Mathematical induction , it is treu for all positive integer n.
(b)Hence, or otherwise, find the values of the following expressions.
(1) 51x52 + 52x53 + 53x54 +...+ 100x101
Solution :
51x52 + 52x53 + 53x54 +...+ 100x101
= (1x2+2x3+...+100x101)-(1x2+2x3+...+50x51)
=(1/3)(100)(101)(102)-(1/3)(50)(51)(52)
=299200//
(2) 1 + (1+2) + (1+2+3) +...+ (1+2+3+...+100)
Solution :
1 + (1+2) + (1+2+3) +...+ (1+2+3+...+100)
=1+(1/2)(2)(1+2)+(1/2)(3)(3+1)+...+(1/2)(100)(100+1)
=(1/2)(1x2+2x3+3x4+...+100x101)
=(1/2)(1/3)(100)(101)(102)
=171700//