已知数列｛an｝满足2an/an+2=an+1(n属于正整数）,a1=1/1006.求证：数列｛1/an｝是等差数列,并求通项an

已知数列｛an｝满足2an/an+2=an+1(n属于正整数）,a1=1/1006.求证：数列｛1/an｝是等差数列,并求通项an
已知数列｛an｝满足2an/an+2=an+1(n属于正整数）,a1=1/1006.求证：数列｛1/an｝是等差数列,并求通项an

已知数列｛an｝满足2an/an+2=an+1(n属于正整数）,a1=1/1006.求证：数列｛1/an｝是等差数列,并求通项an
2an/(an+2)=a(n+1)
1/a(n+1)=(an+2)/(2an)=1/2+1/an
1/a(n+1)-1/an=1/2
所以数列｛1/an｝是等差数列首项为1/a1=1006,公差d＝1/2
则1/an=1006+(n-1)/2=(n+2011)/2
所以an=2/(n+2011)

an+2/2an=1/a(n+1)
1/a(n+1)=1/2+1/an
SO数列｛1/an｝是等差数列首项为1/a1=1006，公差d＝1/2