how to find answer with easy way 1*2+2*3+3*4+4*5...74*75=?

The problem can be reduced to this summation formula:

∑[n(n +1), n=1, 75]=146,300

In Guest #1's reply the upper limit should be 74 rather than 75.  This makes the sum 140600.

 

The sum can also be expressed as ∑[n(n +1) → ∑n² + ∑n  

 

∑n² for n = 1 to N is given by N(N+1)(2N+1)/6

and

∑n  for n = 1 to N is given by N(N+1)/2

 

Using N = 74, these can be used to find the overall sum of 140600.

.