作业帮已知两等差数列an.bn,且a1+a2+.+an/b1+b2+.+bn=3n+1/4n+3,对于任意正整数n都成立,求a
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已知两等差数列an.bn,且a1+a2+.+an/b1+b2+.+bn=3n+1/4n+3,对于任意正整数n都成立,求an：bn.
设{an}的首项为a、公差为A；{bn}的首项为b,公差为B.
[a₁+ a₂+ a₃+ a₄+ .+ an]/[b₁+ b₂+ b₃+ b₄+ .+ bn ]
= [(a+an)n/2]/[(b+bn)n/2]
= (a+an)/(b+bn)
= [a+(n-1)A]/[b+(n-1)B]
[a+(n-1)A]/[b+(n-1)B] = (3n+1)/(4n+3)
当n=1时,a/b = 4/7,a = 4b/7
当n⟼∞,A/B = 3/4
当n=2时,[a+A]/[b+B] = 7/11
11a + 11A = 7b + 7B
11(4b/7) + 11A = 7b + 7(4A/3)
44b/7 + 11A = 7b + 28A/3
A = 3b/7
B = 4A/3 = 4b/7 = a
所以,an = a + (n-1)A = a + (n-1)3b/7 = a + 3(n-1)a/4 = ¼(3n+1)a
任意给定一个a,即可构成{an}
bn = b + (n-1) = 7a/4 + (n-1)B = 7a/4 + (n-1)a = ¼(4n+3)a
对应于a,就可以形成一个符合题意的{bn}.
[a₁+ a₂+ a₃+ a₄+ .+ an]/[b₁+ b₂+ b₃+ b₄+ .+ bn ]
= [(a+an)n/2]/[(b+bn)n/2]
= (a+an)/(b+bn)
= [a+(n-1)A]/[b+(n-1)B]
[a+(n-1)A]/[b+(n-1)B] = (3n+1)/(4n+3)
当n=1时,a/b = 4/7,a = 4b/7
当n⟼∞,A/B = 3/4
当n=2时,[a+A]/[b+B] = 7/11
11a + 11A = 7b + 7B
11(4b/7) + 11A = 7b + 7(4A/3)
44b/7 + 11A = 7b + 28A/3
A = 3b/7
B = 4A/3 = 4b/7 = a
所以,an = a + (n-1)A = a + (n-1)3b/7 = a + 3(n-1)a/4 = ¼(3n+1)a
任意给定一个a,即可构成{an}
bn = b + (n-1) = 7a/4 + (n-1)B = 7a/4 + (n-1)a = ¼(4n+3)a
对应于a,就可以形成一个符合题意的{bn}.
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