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I have the series 2, 2, 6, 10, 22, 42, 86, 170, 342, 682 which follows the following:

M(1)=2

M(2)=2

M(3)=M(1)+M(2)+2

M(4)=M(3)+M(2)+M(1)

M(5)=M(4)+M(3)+M(2)+M(1)+2

M(6)=M(5)+M(4)+M(3)+M(2)+M(1)

Each value for M(N) for N >= 3 is the sum of all previous values, and add 2 if N is odd.

I found that these two functions describe the sum:

[tex]\mbox{If N is odd: }\frac{2^{n+1}+2}{3}[/tex]

[tex]\mbox{If N is even: }\frac{2^{n+1}-2}{3}[/tex]

Can anyone help me inductively prove this?

EDIT: I guess I tried to represent it as a function and failed. If someone can represent this as a summation for me, then I could probably figure out the inductive proof. I guess my whole problem was I tried to represent it as a recursive function, and since the function was wrong, I could not prove it.

M(1)=2

M(2)=2

M(3)=M(1)+M(2)+2

M(4)=M(3)+M(2)+M(1)

M(5)=M(4)+M(3)+M(2)+M(1)+2

M(6)=M(5)+M(4)+M(3)+M(2)+M(1)

Each value for M(N) for N >= 3 is the sum of all previous values, and add 2 if N is odd.

I found that these two functions describe the sum:

[tex]\mbox{If N is odd: }\frac{2^{n+1}+2}{3}[/tex]

[tex]\mbox{If N is even: }\frac{2^{n+1}-2}{3}[/tex]

Can anyone help me inductively prove this?

EDIT: I guess I tried to represent it as a function and failed. If someone can represent this as a summation for me, then I could probably figure out the inductive proof. I guess my whole problem was I tried to represent it as a recursive function, and since the function was wrong, I could not prove it.

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