數學歸納法題

(a)
For n=1,
n(n+1)=1(2)=2
The statement is true for n=1
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Assume the statement is true for n=k
i.e. k(k+1)=2M, where k,M are positive integers
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For n=k+1
(k+1)(k+2)
=k(k+1)+2(k+1)
=2M+2(k+1)
=2(M+k+1)
The statement is true for n=k+1
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By the principle of Mathematical Induction,
the statement is true for all positive integers n.
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(b)
For n=1,
n^3+5n=1+5=6
The statement is true for n=1
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Assume the statement is true for n=k
i.e. k^3+5k=6N, where k,N are positive integers
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For n=k+1
(k+1)^3+5(k+1)
=k^3+3k^2+3k+1+5k+5
=(k^3+5k)+3k^2+3k+6
=6N+3k(k+1)+6
=6(N+1)+3k(k+1)
Since
k(k+1)=2M (proved)
Therefore
6(N+1)+3k(k+1)
=6(N+1)+6M
=6(M+N+1)
The statement is true for n=k+1
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By the principle of Mathematical Induction,
the statement is true for all positive integers n.

數學歸納法 wo, use MI prove

i only can answer no.1

if N is even, then N+1 is odd
even*odd=even

if N is odd, then N+1 is even
odd*even=even