Numbers and Functions (20 marks) Urgent!!!!!!!!!!!!!!!!!!!

1)
If N is odd integer,
(-1)^N + (N+1) + (-1)^2N - (N-1)
= (-1) + (N+1) + 1 - (N-1)
= -1 + N + 1 + 1 - N + 1
= 2
If N is even integer,
(-1)^N + (N+1) + (-1)^2N - (N-1)
= 1 + N + 1 + 1 - N + 1
= 4

2)
Let 2x-1 be the smaller odd number, then the next consecutive odd number is 2x+1
(2x+1)^2 - (2x-1)^2
= (2x+1-2x+1)(2x+1+2x-1)
= (2)(4x)
= 8x
Since 8x is a multiple of 8, the difference between the squares of any two consecutive odd numbers is a multiple of 8

2007-09-28 15:59:15 補充：
I misunderstood the Q1, redo...(-1)^[N (N 1)] (-1)^[2N-(N-1)]= (-1)^(2N 1) (-1)^(N 1) = (-1)^(N 1)[(-1)^N 1]Case 1: N is odd number = 1 [(-1) 1] = 0Case 2: N is even number = (-1) [1 1] = -2

1. Given that N is an integer, find the values of the following expression.
(-1)^[N+(N+1)] + (-1)^[2N-(N-1)]
N+(N+1) = 2N + 1 = odd no.
2N-(N-1) = N + 1 可單可雙, depend on N.
真係唔 sure, 你個 expression 係點架?
唔清唔楚,點幫你呀?

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2. Prove that the difference between the squares of any two consecutive odd numbers is a multiple of 8.
(2n+1)^2 - (2n-1)^2
= 4n^2 + 4n + 1 - (4n^2 - 4n + 1)
= 8n
which is a multiple of 8