各项和为正数的数列an和bn满足an,bn,an+1成等差数列,bn,an+1,bn+1成等比数列 求证(根号bn)是等
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各项和为正数的数列an和bn满足an,bn,an+1成等差数列,bn,an+1,bn+1成等比数列 求证(根号bn)是等差数列
题意:an+an+1=2bn; (1)
bnbn+1=an+1*an+1 (2)
(2)式两边开方得:an+1=sqrt(bn)*sqrt(bn+1) (3)
(1)式两边平方,展开,然后将(3)式代入,可得:
bn*bn-1+bn*bn+1+2*sqrt(bn-1*bn*bn*bn+1)=4bn*bn (4)
整理(4)式后可得
sqrt(bn-1)+sqrt(bn+1)=2sqrt(bn)
bnbn+1=an+1*an+1 (2)
(2)式两边开方得:an+1=sqrt(bn)*sqrt(bn+1) (3)
(1)式两边平方,展开,然后将(3)式代入,可得:
bn*bn-1+bn*bn+1+2*sqrt(bn-1*bn*bn*bn+1)=4bn*bn (4)
整理(4)式后可得
sqrt(bn-1)+sqrt(bn+1)=2sqrt(bn)
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