Can anyone please help? I don't know if I am doing it right. I am stuck in the inductive step.

Using induction please prove the following formula:

u2 + 2 u4 + 3u6 +...+ n u2n = n u(2n+1) - u2n

where the numbers to the the right of the letter u are subscripts.

So far I proved

Base Case: n = 1

LHS: 1 *u2(1) = 1*u2 = 1*1 =1

RHS: 1* u(2(1)+1) - u2(1) = 1*u3 -u2 = u3-u2 = 2-1 =1

I.H.= Assume for all n to be true

u2 + 2 u4 + 3u6 +... + n u2n = n u(2n+1) - u2n

I.S. Prove for n+1

u2 +2 u4+3 u6+.. +n u2n + (n+1) u2(n+1) = (n+1) u(2(n+1)+1) - u2(n+1)

By I.H. then,

n u(2n+1) - U2n + (n+1) u2(n+1)

and then I get stuck....