Question: 1.Use induction to prove: For every integer n 1, 1*2 + 2*3 + 3*4 + ... + n(n+1) = [n(n+1)(n+2)]/3. 2.Here is a proof that

1.Use induction to prove: For every integer n 1,

1*2 + 2*3 + 3*4 + ... + n(n+1) = [n(n+1)(n+2)]/3.

2.Here is a proof that for n 0, 1 + 2 + 22 + + 2n = 2n+1.

Proof. Suppose 1 + 2 + 22 + + 2n = 2n+1 for some n 0. Then

1 + 2 + 22 + + 2n + 2n+1 = 2n+1 + 2n+1using the inductive hypothesis

= 2(2n+1) = 2n+2 = 2(n+1)+1,

as we needed to show.

  1. Now, obviously there is something wrong with this proof by induction since, for example, 1 + 2 + 22 = 7, but 22+1 = 23 = 8. What specifically is wrong with the proof?

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