數學 6題M.I and 1題binomial~~~~急問

1.show that for any positive interger n,there exist positive integers An and Bn such that (1+√3)^2n = An+Bn√3 where (An)^2 -3(Bn)^2=2^2n and An and Bn are both divisible by 2^n.

2.The Fibonacci Sequence { Fn} is defined as follows:
F1=F2=1:Fn+2=Fn+1+Fn where n is any natural number.
Prove that for any natural numbers n and k,where k >=3,Fkn is a multiple of Fk.
Hint:Show that Fkn=(Fs)(Fkn-s+1)+(Fs-1)(Fkn-s) where 2=<s=<k

3.Prove that 1/√(2n+1) >(1x3x5...(2n-1)) / (2x4x6...(2n)) >√(n+1) / (2n+1)

4.Prove that the cubes of the natural numbers beginning with 1 leave,when divided by 6,the remainder 1,2,3,4,5,0 recurring in order.

5.Prove that any n squares can be cut and pasted into one large square.

6.Prove that the maximum possible number of regions that a space can be divided by n planes is 1+(1/6)n(n^2 +5).

7.Prove that m+nCr+1=mCr+1 + mCr + m+1Cr + m+2Cr +...+ m+n-1Cr