题目

New evidence shows that getting enough Vitamin D may be the most important thing you can do for your health． You know the usual prescription for good health: a balanced diet with lots of fruits and vegetables, regular exercise, no smoking．Now add this: Spend a little more time in the sun．     Huh? That may sound like medical heresy （反面观点）．After all, we've been warned for decades about the dangers of the sun: wrinkles, age spots and the increasing threat of skin cancer．But new and impressive medical evidence suggests that sunlight is beneficial．The vitamin D it promp（2013•广东）设各项均为正数的数列{an}的前n项和为Sn，满足4Sn=a2n+1-4n-1，n∈N*，且a2，a5，a14构成等比数列．（1）证明：a2=4a1+5；（2）求数列{an}的通项公式；（3）证明：对一切正整数n，有1a1a2+1a2a3+…+1anan+1＜12．