Prove that `1+1* ""^(1)P_(1)+2* ""^(2)P_(2)+3* ""^(3)P_(3) + … +n* ""^(n)P_(n)=""^(n+1)P_(n+1).`

Explanation

`LHS=.^(1)P_(1)+2*.^(2)P_(2)+3*.^(3)P_(3)+ . . .+n*.^(n)P_(n)`
`=underset(r=1)overset(n)(sum)r*.^(r)P_(r)=underset(r=1)overset(n)(sum){(r+1)-1}*.^(r)P_(r)`
`=underset(r=1)overset(n)(sum){(r+1)*.^(r)P_(r)-.^(r)P_(r))}`
`=underset(r=1)overset(n)(sum)(.^(r+1)P_(r+1)-.^(r)P_(r))` [from note (iii)]
`=.^(n+1)P_(n+1)-.^(1)P_(1)=.^(n+1)P_(n+1)-1`
`=RHS`