数列an满足a1=1/4,a2=1/5,且a1a2+a2a3+...+anan+1数列{an}满足a1=1/4,a2=1/5,且a1a2+a2a3+…+anan+1=na1an+1对于任何正整数n都成立,则1/a1+1/a2+...+1/a97的值为
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![数列an满足a1=1/4,a2=1/5,且a1a2+a2a3+...+anan+1数列{an}满足a1=1/4,a2=1/5,且a1a2+a2a3+…+anan+1=na1an+1对于任何正整数n都成立,则1/a1+1/a2+...+1/a97的值为](/uploads/image/z/7119217-1-7.jpg?t=%E6%95%B0%E5%88%97an%E6%BB%A1%E8%B6%B3a1%3D1%2F4%2Ca2%3D1%2F5%2C%E4%B8%94a1a2%2Ba2a3%2B...%2Banan%2B1%E6%95%B0%E5%88%97%7Ban%7D%E6%BB%A1%E8%B6%B3a1%3D1%2F4%2Ca2%3D1%2F5%2C%E4%B8%94a1a2%2Ba2a3%2B%E2%80%A6%2Banan%2B1%3Dna1an%2B1%E5%AF%B9%E4%BA%8E%E4%BB%BB%E4%BD%95%E6%AD%A3%E6%95%B4%E6%95%B0n%E9%83%BD%E6%88%90%E7%AB%8B%2C%E5%88%991%2Fa1%2B1%2Fa2%2B...%2B1%2Fa97%E7%9A%84%E5%80%BC%E4%B8%BA)
数列an满足a1=1/4,a2=1/5,且a1a2+a2a3+...+anan+1数列{an}满足a1=1/4,a2=1/5,且a1a2+a2a3+…+anan+1=na1an+1对于任何正整数n都成立,则1/a1+1/a2+...+1/a97的值为
数列an满足a1=1/4,a2=1/5,且a1a2+a2a3+...+anan+1
数列{an}满足a1=1/4,a2=1/5,且a1a2+a2a3+…+anan+1=na1an+1对于任何正整数n都成立,则1/a1+1/a2+...+1/a97的值为
数列an满足a1=1/4,a2=1/5,且a1a2+a2a3+...+anan+1数列{an}满足a1=1/4,a2=1/5,且a1a2+a2a3+…+anan+1=na1an+1对于任何正整数n都成立,则1/a1+1/a2+...+1/a97的值为
a1a2+a2a3+…+ana(n+1)=na1a(n+1)
a1a2+a2a3+…+a(n-1)ana=(n-1)a1an
两式相减得
ana(n+1)=na1a(n+1)-(n-1)a1an
等式两边同时除以ana(n+1)
1=na1/an-(n-1)a1/a(n+1)
1/a1=n/an-(n-1)/a(n+1)
n/an-(n-1)/a(n+1)=1/(1/4)
n/an-(n-1)/a(n+1)=4.1
同理得
(n-1)/a(n-1)-(n-2)/an=4.2
1式-2式得
2(n-1)/an-(n-1)/a(n+1)-(n-1)/a(n-1)=0
2(n-1)/an=(n-1)/a(n+1)+(n-1)/a(n-1)
2/an=1/a(n+1)+1/a(n-1)
所以1/an是等差数列.
d=1/a2-1/a1
=1/(1/5)-1/(1/4)
=5-4
=1
1/an=1/a1+(n-1)d
=1/(1/4)+n-1
=4+n-1
=n+3
1/a1+1/a2+...+1/a97
=4+5+.+100
=(4+100)*97/2
=5044