Question

Prove that 1 + 1. 1p1 + 2. 2p2 + 3. 3p3 + . N. Npn = n+1pn+1

Answer 1

1 + 1. 1p1 + 2. 2p2 + 3. 3p3 + ...... N. Npn = n+1pn+1

We know that npn = n ! / ( n - n) ! = n !

We can prove using Principle of Mathematical induction

for  n = 2

• 1 + 1 . 1p1 = 2 = 2 ! = 2p2

• 1 + 1 . 1p1 + 2 . 2p2 = 1 + 1 + 2 . 2! = 2 + 4 = 6 = 3 ! = 3p3

Therefore lets assume its true for n = k

• 1 + 1. 1p1 + 2. 2p2 + 3. 3p3 + ...... k. kpk = k+1pk+1  -- ( 1 )

For n = k + 1 ,  

• 1 + 1. 1p1 + 2. 2p2 + 3. 3p3 + .. k. kpk + ( k + 1 ) .k+1pk+1  = k + 2pk + 2 - - ( 2 )

( 2 ) - ( 1 ) = =>

• LHS  =  ( k + 1 )( k + 1 ) !

• RHS = ( k + 2 ) ! - ( k + 1 ) ! = ( k + 1 ) ! ( k + 2 - 1 ) = ( k + 1 ) (  k + 1 ) !

• Since RHS = LHS , The expression is true for all n.