Question

已知数列{an}满足：a1=1,a(n+1下标)=an/an+1

Answer 1

1.

a(n+1)=an/(an +1)

1/a(n+1)=(an +1)/an =1/an +1

1/a(n+1)-1/an=1,为定值.

1/a1=1/1=1

数列{1/an}是以1为首项,1为公差的等差数列.

1/an=1+(n-1)=n

an=1/n

数列{an}的通项公式为an=1/n.

2.

2^n/an=2^n/(1/n)=n×2^n

Sn=1×2+2×2^2+3×2^3+...+n×2^n

2Sn=1×2^2+2×2^3+...+(n-1)×2^n+n×2^(n+1)

Sn-2Sn=-Sn=2+2^2+...+2n -n×2^(n+1)=2×(2^n -1)/(2-1)-n×2^(n+1)=(1-n)×2^(n+1) -2

Sn=(n-1)×2^(n+1) +2