Question

注:一定要用完整的数学归纳法解题

Answer 1

1.

当n=2时,

1/3+1/4+1/5+1/6=57/60=19/20>5/6

设当n=k时,不等式成立

1/(k+1)+1/(k+2)+……+1/3k>5/6

当n=k+1时

1/(k+1+1)+1/(k+1+2)+……+1/3k+1/(3k+1)+1/(3k+2)+1/(3k+3)

=1/(k+1)+1/(k+2)+……+1/3k+[1/(3k+1)+1/(3k+2)+1/(3k+3)-1/(k+1))]

1/(k+1)+1/(k+2)+……+1/3k>5/6

1/(3k+1)+1/(3k+2)+1/(3k+3)-1/(k+1)>0

(因为1/(k+1)=3*1/(3k+3))

所以当n=k+1时

1/(k+1+1)+1/(k+1+2)+……+1/3k+1/(3k+1)+1/(3k+2)+1/(3k+3)>5/6

2.

当n=5时

2^5=32>5^2

设当n=k时成立,2^k>k^2

当n=k+1时

2^(k+1)=2^k+2^k>2*k^2

(k+1)^2=k^2+2k+1(k+1)^2