Manipulation of sums: perturbation

[slapstick4235]

Can someone explain to me step by step how we can go from this
S_n+(n+1)2ⁿ⁺¹=2S_n+2ⁿ⁺²-2
To this?
S_n=(n-1)2ⁿ⁺¹+2

 

[Ishuda]

Can someone explain to me step by step how we can go from this
S_n+(n+1)2ⁿ⁺¹=2S_n+2ⁿ⁺²-2
To this?
S_n=(n-1)2ⁿ⁺¹+2

Can you subtract S_n from each side, then subtract 2ⁿ⁺² - 2 = 2 (2ⁿ⁺¹ - 1) from each side and simplify?

 

[pka]

Can someone explain to me step by step how we can go from this
S_n+(n+1)2ⁿ⁺¹=2S_n+2ⁿ⁺²-2
To this?
S_n=(n-1)2ⁿ⁺¹+2

\(\displaystyle S_n+(n+1)2^{n+1}=2S_n+2^{n+2}-2\\(n+1)2^{n+1}-2^{n+2}+2=S_n\\2^{n+1}[(n+1)-2]+2=S_n\\(n-1)2^{n+1}+2=S_n\)

 

[slapstick4235]

Thank you pka and Ishuda. I understand now.