无穷数列an中,a1=1,an=√(an-1)^2+4,(n>=2,n属于N*)已知数列{an}中,a1=1,an=√a(n-1)^2+4.(n≥2,n属于N*)（1）求{an}的通项公式（2）设bn=1/an+a（n+1）,求｛bn｝的前100项和（3）求lim（a(n+2)-a(n+1))/(a(n+1)-an)

Question

无穷数列an中,a1=1,an=√(an-1)^2+4,(n>=2,n属于N*)
已知数列{an}中,a1=1,an=√a(n-1)^2+4.(n≥2,n属于N*)（1）求{an}的通项公式（2）设bn=1/an+a（n+1）,求 ｛bn｝的前100项和
（3）求lim（a(n+2)-a(n+1))/(a(n+1)-an)

Answer 1

(1)
an=√[(a(n-1))^2+4]
(an)^2 - (a(n-1))^2=4
{ (an)^2} 是等差数列, d=4
(an)^2 - (a1)^2=4(n-1)
(an)^2 = 4n-3
an = √(4n-3)
(2)
bn = 1/(an +a(n+1) )
=1/(√(4n-3) + √(4n+1) )
= [√(4n+1)-√(4n-3) ] / 4
b1+b2+...+b100
= [√(4(100)+1)-√(4-3) ] / 4
= [√401-1 ] / 4
(3)
lim(n->∞)（a(n+2)-a(n+1))/(a(n+1)-an)
=lim(n->∞)[ √(4n+4) -√(4n+1)]/[√(4n+1)-√(4n-3) ]
consider
lim(x->∞)[ √(4x+4) -√(4x+1)]/[√(4x+1)-√(4x-3) ]
=lim(x->∞)[ √(4+4/x) -√(4+1/x)]/[√(4+1/x)-√(4-3/x) ] (0/0)
=lim(x->∞){ (-4/[2x^2.√(4+4/x)]) +1/[2x^2.√(4+1/x)] }/{-1/[2x^2.(4+1/x)]-3/[2x^2.√(4-3/x) ]}
=lim(x->∞){ (-4/[2.√(4+4/x)]) +1/[2.√(4+1/x)] }/{-1/[2.√(4+1/x)]-3/[2.√(4-3/x) ]}
=( -1+1)/(-1/4-3/4)
=0
lim(n->∞)（a(n+2)-a(n+1))/(a(n+1)-an)
=lim(n->∞)[ √(4n+4) -√(4n+1)]/[√(4n+1)-√(4n-3) ]
=0