Chứng minh rằng các đẳng thức sau đúng với mọi n thuộc N*

Hướng dẫn giải

a) 1+2Cn1+4Cn2+…+2n−1Cnn−1+2nCnn

=Cn01+Cn12+Cn222+…+Cnn−12n−1+Cnn2n

=Cn01n+Cn11n−12+Cn21n−222+…+Cnn−11 .2n−1+Cnn2n

= (1 + 2)n = 3n.

b) Ta có:

(x+1)2n=C2n0x2n+C2n1x2n−11+C2n2x2n−212+…+C2n2n−1x12n−1+C2n2n12n
=C2n0x2n+C2n1x2n−1+C2n2x2n−2+…+C2n2n−1x+C2n2n.

Cho x = –1, ta được:

(−1+1)2n=C2n0(−1)2n+C2n1(−1)2n−1+C2n2(−1)2n−2+…+C2n2n−1(−1)+C2n2n

=C2n0−C2n1+C2n2−…−C2n2n−1+C2n2n

 

⇒C2n0−C2n1+C2n2−…−C2n2n−1+C2n2n=0
⇒C2n0+C2n2+C2n4+…+C2n2n=C2n1+C2n3+C2n5+…+C2n2n−1.