For any positive integers m, n (with `n ge m`)
If `({:(n),(m):}) = .^(n)C_(m)` Prove that
`({:(n),(m):}) + ({:(n - 1),(m):}) + ({:(n - 2),(m):}) + … + ({:(m),(m):}) = ({:(n + 1),(m + 1):})`
Prove that
`({:(n),(m):}) + 2 ({:(n + 1),(m):}) + 3 ({:(n - 2),(m):}) + .... + (n - m + 1)`
`({:(m),(m):}) = ({:(n + 2),(m + 2):})`

Explanation

Let `S = ((n)/(m))+((n-1)/(m)) + ((n-2)/(m))+ ((n-2)/(m)) +.....+((m)/(m)) =((n+1)/(m+1)).....(i)`
It is obvious that, `n ge m`
Note : This question is based upon additive loop.
Now ,`S =((m)/(m)) + ((m+1)/(m)) + ((m+2)/(m)) +.......+((n)/(m))`
` ={((m+1)/(m+1))+((m+1)/(m))}[because ((m)/(m)) = 1= ((m+1)/(m+1))]`
`= ((m+2)/(m+1)) + ((m+2))/(m)) + ......+ ((n)/(m))" "[because""^(n)C_(r)+ ""^(n)C_(r+1) = ""^(n+1)C_(r+1)]`
`=((m+2)/(m+1)) +......+((n)/(m))`
`=.............`
` =((n)/(m+1))+((n)/(m)) = ((n+1)/(m+1))` which is ture ....(ii)
Again, we have to prove that
`((n)/(m))+2((n-1)/(m)) + 3((n-2)/(m)) +......+ (n-m+1)((m)/(m)) = ((m+2)/(m+2))`
Let `S_(1) = ((n)/(m))+2((n-1)/(m)) +3((n-2)/(m)) +......+(n-m+1)((mm)/(m))`
`{:(= ((n)/(m)) + ((n-1)/(m)) + ((n-2)/(m)) +...+ ((m)/(m))), ( " "+ ((n-1)/(m)) + ((n-2)/(m)) +...+ ((m)/(m))), (" "+ ((n-2)/(m)) +...+((m)/(m)) ),(" "...) , (" "+ ((m)/(m))):}}n-m + 1` rows
Now, sum of the first row is `((n+1)/(m+1))`
Sum of the second row is `((n)/(m+1))`
Sum of the third row is `((n+1)/(m+1))`,
....................
Sum of the last row is `((m)/(m)) = ((m+1)/(m+1))`
Thus `S = ((n+1)/(m+1))+((n)/(m+1)) + ((n +1)/(m+1))+.......+ ((m+1)/(m+1)) = ((n+1+1)/(m+2)) = ((n+2)/(m+2))`
[from Eq. (i) replacing n by n +1 and m by m + 1]