Proof by induction

u_n = 3 * 2^n-1 - 1     

Prove this is true for  n = 1

u₁ = 3 * 2 ^{1 - 1} + 1

u₁ = 3 * 1 - 1 =   2     which is true

Asume that it is true for n = k

That is : 

u_k = 3 * 2^k-1 - 1

Show that it is true for n = k + 1

u_k+1 = 3 * 2 ^{(k + 1 - 1)}  - 1

u_k+1 = 3 * 2^k - 1     (1)

And we were given  that

u_k+1  = 2u_k + 1        (2)

So....by our assumption u_k = 3 * 2^k-1 - 1    (3)

Sub    (3) into (2)

u_k+1  = 2 ( 3 * 2^k-1 - 1) + 1  =

2 * 3 * 2^k-1 - 2 + 1  =

3 * 2 * 2^k-1 - 1 =

3 * ( 2¹ * 2^k-1) - 1 =

3 * ( 2 ^{k - 1 + 1})  - 1   = 

3 * 2^k - 1        which   equals  ( 1)