Question

高中数学题，求完整解答过程！

Answer 1

21 (1) 过O做AnBn的垂线交于Hn，OHn=hn;a1:a2=1:2,则h1:h2=1:2
AnBn=cn,所以A1B1=A2B2/2=c1，S-A1B1B2A2=S1=3c1(h2-h1)/2=3c1h1/2=(2c1+c3)(h3-2h1)/2
a2/a3=h2/h3=c2/c3,h3=h2c3/c2
所以3c1h1=(2c1+c3)(h2c3/c2-2h1)=(2c1+c3)(h1c3/c1-2h1)=h1(2c1+c3)(c3-2c1)/c1
=(c3^2-4c1^2)h1/c1,
所以c3^2=7c1^2,c3=√7c1,a3/a2=c3/c2=√7/2,a3=√7,h3=√7h1
则3c1h1=(√7c1+c4)(h3c4/c3-√7h1)=(√7c1+c4)(h1c4/c1-√7h1)=(c4^2-7c1^2)h1/c1
所以10c1^2=c4^2,c4=√10c1,a4=√10,h4=√10h1
可以预见有：3c1h1=(c5^2-10c1^2)h1/c1，c5=√13c1,a5=√13,h5=√13h1
所以an的平方值是个公差为3的等差数列，即an=√[1+3(n-1)]=√(3n-2)