徒步自由行

1.

(1)證明C(n,0)+ 3C (n,1)+ 5C (n,2)+...+(2n+1)C(n,n)=(n+1)2^n

Σ(k=1 to n) (2k+1)*C(n,k)

=Σ(k=1 to n) (2k)*n!/[k!*(n-k)!] +Σ(k=1 to n) C(n,k)

=2nΣ(k=1 to n) (n-1)!/[(k-1)!*(n-k)!] +Σ(k=1 to n) C(n,k)

=2nΣ(k=1 to n) C(n-1,k-1) +Σ(k=1 to n) C(n,k)

=2n *2^(n-1) +2^n

=(n+1)*2^n

(2)∑(k=0~n)(3k-2)*C(n,k)=?

=∑(k=1~n)(3k)*C(n,k) - ∑(k=0~n)C(n,k)

=∑(k=1~n)(3n)*C(n-,k-1) - ∑(k=0~n)C(n,k)

= (3n)*2^(n-1) – 2^n

=(3n-2)*2^(n-1)

2.

利用C(n,1)+ 2C (n,2)+ 3C (n,3)+...+n(C(n,n))=n*2^(n-1)

(1)證明C(n,1)*C(n,2)*C(n,3)*....*C(n,n)<2^[n(n-1)]/n!

算幾：

[C(n,1)+ 2C (n,2)+...+n(C(n,n))]/n > [n!* C(n,1)*C(n,2)*C(n,3)*....*C(n,n)]^(1/n)

[n!* C(n,1)*C(n,2)*C(n,3)*....*C(n,n)]^(1/n) < 2^(n-1)

n!* C(n,1)*C(n,2)*C(n,3)*....*C(n,n) < 2^[n(n-1)]

C(n,1)*C(n,2)*C(n,3)*....*C(n,n) < 2^[n(n-1)]/n!

(2) 2C (n,0)+ 5C (n,1)+ 8C (n,2)+...+(3n+2)C(n,n)=?

 Σ(k=0 to n) (3k+2) C(n,k)

=3Σ(k=0 to n) k C(n,k)+2Σ(k=0 to n) C(n,k)

=3Σ(k=1 to n) n C(n-1,k-1)+2*2^n

=3 n *2^(n-1) +2*2^n

=(3 n +4)*2^(n-1)

3.

證明C(n,0)+1/ 3C (n,2)+1/ 5C (n,3)....=2^n/n+1

4.

利用恆等式(a+b)^3(a+b)^6=(a+b)^9

(1)證明C(3,0)*C(6,4)+C(3,1)*C(6,3)+C(3,2)*C(6,2)+C(3,3)*C(6,1)=C(9,4)

(a+b)^3=C(3,0)*a^3+C(3,1)a^2 b +…..+C(3,3)b^3

(a+b)^6=C(6,0)*a^6+C(6,1)a^6 b +….. +C(6,4)a^2b^4+C(6,5)ab^6+C(6,6)b^6

兩式相乘

a^5b^4 的系數= C(3,0)*C(6,4)+C(3,1)*C(6,3)+C(3,2)*C(6,2)+C(3,3)*C(6,1)

等右式 C(9,4)

(2)證明C(m,0)*C(n,0)+C(m,1)*C(n,1)+C(m,2)C(n,2)+...+C(m,m)*C(n,m)=(n+m)!/m!n!

(1+x)^m=C(m,0)x^m+C(m,1)x^(m-1)+………+C(m,m)

(1+x)^n=C(n,0)+C(n,1)x+……………….……+C(n,m)x^m+……+C(n,n)

兩式相乘 (1+x)^(m+n)

x^m 系數=

C(m,0) C(n,0)+ C(m,1) C(n,1)+…………+C(m,m)C(n,m)

=C(m+n ,m)=(m+n)!/[m!*n!]

(3)C(10,0)*C(10,2)+C(10,1)*C(10,3)+...+C(10,8)*C(10,10)

(1+x)^10= C(10,0)x^10+C(10,1)x^9+………+C(10,10)

(1+x)^10= C(10,0)+C(10,1)x+C(10,2)x^2………+C(10,10)x^10

兩式相乘 (1+x)^20

x^12 系數= C(10,0)*C(10,2)+C(10,1)*C(10,3)+...+C(10,8)*C(10,10)

=C(20,12)