(1)在数列{an},a1=2,an=2an-1+2n+1(n≥2,n∈N*),令{bn}= an/2n,求证：{bn}是等差数列；(2)在(1)(1)在数列{an},a1=2,an=2a^(n-1)+2^(n+1)(n≥2,n∈N*),令{bn}= a^n/2^n,求证：{bn}是等差数列；(2)在(1)的条件下,设T^n=1/b1b2+1b2b3+…+bnbn+1,求T^n
问题描述：
(1)在数列{an},a1=2,an=2an-1+2n+1(n≥2,n∈N*),令{bn}= an/2n,求证：{bn}是等差数列；(2)在(1)
(1)在数列{an},a1=2,an=2a^(n-1)+2^(n+1)(n≥2,n∈N*),令{bn}= a^n/2^n,求证：{bn}是等差数列；
(2)在(1)的条件下,设T^n=1/b1b2+1b2b3+…+bnbn+1,求T^n
答
1,an=2a(n-1)+2^(n+1),两边除以2^n,得an/2^n=a(n-1)/2^(n-1)+2,即bn=b(n-1)+2,所以{ bn }是等差数列,b1=a1/2=1,bn=1+(n-1)*2=2n-1；2,1/bnb(n+1)=1/(2n-1)(2n+1)=1/2*[1/(2n-1)-1/(2n+1)],所以Tn=1/b1b2 +1/ b2b3 +....