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The procedure of proof should follow the following format. The answer should like that. Example Consider the sequence a1, a2, a3,... given by a1= 2,

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The procedure of proof should follow the following format. The answer should like that.

Example

Consider the sequence a1, a2, a3,... given by a1= 2, a2= 5, and ai= ai-1+ 2ai-2for all i3.

Prove that, for all i1:ai= (1/3)(72i-1+ (-1)i)

Proof: We prove the statement by induction on i.

Base Case:

We prove that ai= (1/3)(72i-1+ (-1)i) when i=1 and i=2.

When i=1: we are given that a1= 2,and we see that (1/3)(72i-1+ (-1)i) = (1/3)(7-1) = 2, as required.

When i=2: we are given that a2= 5,and we see that (1/3)(72i-1+ (-1)i) = (1/3)(72+1) = (1/3)15 = 5, as require

Inductive step: We prove that, for all k2,if aj= (1/3)(72j-1+ (-1)j) for all j{1,...,k},then ak+1= (1/3)(72k+ (-1)k+1).

(1)Let k be an arbitrary positive integer such that k2.Assume thataj= (1/3)(72j-1+ (-1)j) for all j{1,...,k}.

(2)Since k2, we know that k+13, so the given recursive formula states that ak+1= ak+ 2ak-1.

(3)From (1), we know that ak= (1/3)(72k-1+ (-1)k) and ak-1= (1/3)(72k-2+ (-1)k-1).

(4)From (2) and (3), it follows that ak+1= (1/3)(72k-1+ (-1)k) + 2(1/3)(72k-2+ (-1)k-1)= (1/3)(72k-1+ (-1)k) +(1/3)(72k-1+ 2(-1)k-1)= (1/3)(72k-1+ (-1)k) +(1/3)(72k-1-2(-1)k)= (1/3)(72k-1+ 72k-1+ (-1)k- 2(-1)k)= (1/3)(72k- (-1)k) = (1/3)(72k+ (-1)k+1) , which completes the inductive step.

7. Prove by induction on n: for all integers n 0, for every set S containing exactly n elements, [P(S)2

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